HELMUT SCHEUERMANN AND HElNZ GOCKLER - der ...

Reprint from 

PROCEEDINGS 

of the IEEE 69 

November l981 

pp. 1419--1450 

HELMUT SCHEUERMANN 

AND HElNZ GOCKLER

PROCEEDINGS OF THE IEEE, VOL. 69, NO. 11, KWEMBER 1981 

HELMUT SCHEUERMANN AND HEINZ aCKLER 

Abstract-With this survey, an attempt is made to describe the pat 

majority of all known methods of digital transmultiplexing (Le., conversion) 

of time-division-multiplex (TDM) to frequency-divisionmultiplex 

(FDM) signals, and vice versa. To this end, the individual 

transmultiplexer approaches are classified into four categories according 

to the undedying algorithm: Bandpass filter bank, low-pass filter 

bank, Weaver structure method, and multistage modulation method. 

Finally, the overall performance of the various transmultiplexer approaches 

are compared with each other by means of different criteria 

[l], such as stability under looped conditions, absolute value of the 

group delay, computational and control complexity, modularity, potential 

of intelligible crosstalk, absence of an additional analog frequency 

conversion, and the impact of out+£-band signaling. For a more profound 

understanding of the individual digital transmultiplexer approaches, 

the main chapter is preceded by an introductory discussion 

on analog and digital generation of single-sideband signals. In this 

context, the associated problems of sample rate alteration and multirate 

filtering arising f~om digital signal processing are dealt with. 

Manuscript received February 8, 1980; revised July 30, 1981. This 

work was supported in part by the German Federal Ministry of Research 

and Technology. 

The authors are with the Advanced Development Department, AEG- 

Telefunken Kommunikationstechnik, P.O.B. 1120, D-71 50 Backnang, 

Germany. 

I. INTRODUCTION 

A 

NALOG TRANSMISSION and switching facilities for 

telephony signals are nowadays to a growing extent 

being expanded and replaced by digital facilities. 

Thereby the conventional multiple utilization of transmission 

paths. by frequency-division-multiplex (FDM) is being substituted 

by time-division-multiplex (TDM) techniques. The 

chief advantages of digital TDM transmission as compared with 

analog FDM transmission are as follows: 

1) no generation of additive noise on the transmission 

path; 

2) within certain limits, no occurrence of interference 

through crosstalk; 

3) possibility for concentration of switching and transmission 

facilities. 

The widespread use and high investment outlays of the 

installed facilities will require that analog and digital technologies 

coexist well into the foreseeable future. This will lead to 

an increasing extent to interfaces between analog and digital 

sections of the toll communication network. Interconnection

PROCEEDINGS OF THE IEEE, VOL. 69, NO. 11, NOVEMBER 1981 

FDM: 

FDM 

Frequency Division 

Multiplex 

SG : Supergroup 

PDM : Time Divi 

Multiplex 

sion 

Fig. 1. Block diagram of a 60-channel transmultiplexer. (a) TDM-FDM 

direction. (b) FDM-TDM direction. 

at these places could be made, in principle, using the available 

FDM and TDM terminal-station equipment. In doing so, the 

process goes-for instance, when converting from the analog 

to the digital signal representation-from the particular FDMcarrier 

hierarchy stage down to the voice frequency level and 

proceeds from there to the corresponding TDM hierarchy stage. 

Better system performance and a more economical solution 

may be achieved with an interfacing facility specially designed 

for this purpose. For such facility, the designation "transmultiplexer" 

has been coined. 

Signal processing in a transmultiplexer can be done either in 

the analog or in the digital way. As compared with analog 

technology, digital technology offers the advantages of being 

suited for integration, and of the absence of circuit adjustment 

and variation of equipment performance with time and temperature, 

etc. 

A great number of publications on digitally implemented 

transmultiplexers appeared over the last years [ 1 ]-[7]. 

With 

the increasing number of transmultiplexing methods' being 

disclosed, it becomes ever niore difficult to maintain an overview 

and, for given conditions, to choose a suitable approach. 

It is therefore the aim of this contribution to describe the fundamental 

algorithms in a tutorial manner, to point out the 

common features of the individual known transmultiplexer 

approaches and to classify these methods according to the 

inherent digital signal processing procedures. To this end, the 

functions of individual subassemblies of a transmultiplexer are 

explained in the next section. For a deeper understanding of 

digital transmultiplexer algorithms, the possibilities of analog 

single-sideband modulation are recalled and, in addition, extended 

to digital signal processing in the subsequent sections. 

Particularly, the questions of sample rate alteration and multirate 

filtering (interpolation, decimation) are treated in detail. 

In the main section, the individual known transmultiplexer approaches 

are classified into four categories according to the 

underlying algorithm: Bandpass filter-bank method, low-pass 

fiiter-bank method, Weaver structure method, and multistage 

modulation methods. Finally, in the concluding sections, the 

various transmultiplexing methods described in this paper are 

compared with each other on the basis of a set of criteria 

essentially introduced by Fettweis [ 1 1. 

In its basic function, a transmultiplexer represents a singlesideband 

modulator. As an example for the European format, 

Fig. 1 shows the functional block diagram of a 60channel 

transrnultiplexer for the TDM-FDM-direction (Fig. l(a)) and 

for the FDM-TDM-direction (Fig. l(b)): The data streams of 

two PCM TDM systems, each of them comprising 30 telephone 

channels corresponding to the lowest European TDM hierarchical 

level (PCM 30), are simultaneously translated to the 

FDM supergroup (SG) level in the frequency range from 3 12 

kHz to 552 kHz, and vice versa [ 11, [2]. In contrast to that, 

in North America and Japan the lowest hierarchy of the TDM 

system is made up of only 24 telephone channels. Under these 

constraints, a transmultiplexer is generally applied to the conversion 

of one 24channel PCM TDM data stream to two 

(primary) groups of the FDM hierarchy, each of them comprising 

12 channels in the frequency band ranging from 60 to 

108 kHz, and vice versa [ l], [2]. Moreover, in Japan some 

attempts have been made to realize a 120-channel transmultiplexer, 

corresponding to an interface between five 24-channel 

PCM TDM formats and two FDM supergroups [ l], [8], [9]. 

Subsequently, however, the more general discussion will predominantly 

be based on the 60channel approach. Nevertheless, 

North American and Japanese 24-channel transmultiplexers 

(which can usually be adapted to another number of 

channels without difficulty) will be considered in the main 

chapter as well. On the contrary, facilities, such as those required 

for signaling, dialing pulses conversion, and supervision 

will only be treated marginally, since they have no bearing on 

the understanding of the basic principles of operation. The 

interested reader is referred to [75], [76]. 

In the European approach, the two PCM TDM bit streams 

(PCM 30), each with a transmission speed of 2.048 Mbitls, are 

applied to a receiver R (Fig. l(a)). Here, the two TDM signals

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTIPLEXING METHODS 1421 

TABLE I 

equipment are compiled in the CCITT Recommendation 

SOME SPECIFICATIONS FOR 60-CHAXNEL TRANSMULTIPLEXERS ACCORDING G.792. In addition, some general considerations on trans- 

TO CCIn RECOMMENDATION G. 792 AND G. 793 

multiplexers can be taken from the CCITT Recommendation 

Magnitude rcsponsc * 

G.791, whereas the particular specifications of a 60channel 

transmultiplexer, such as pilots, signaling, etc., can be found in 

6WHz to 2LOOHz -0.6dB 5 a 50.6dB 

the CCITT Recommendation G.793. From these recommenda- 

Maximum allowancc at thr band edges 

tions, the most important requirements for a 60channel trans- 

300 Hz -0.6dB r a C 1.7dB multiplexer are given in Table I. It must be noted that these 

3400 Hz -0.6dB 5 o S 2.LdB 

data are, in general, to be measured at the analog (FDM) ports, 

the digital (TDM) ports being looped. As a consequence, the 

Group delay * (absolute value) T C 3ms 

actual constraints to be imposed on any individual filter of a 

transmultiplexer may be much more stringent. Furthermore, 

Distortion 

1000 Hz to 2600Hz AT 4 0Sms 

the individual filter specifications are generally different for 

the various transmultiplexing methods to be described in the 

sequel. 

600 Hz to l000 Hz AT a 1.5n-s 

504 Hz to 600Hz, 2600Hz to 28WHz AT C 2ms 

Minimum crosstalk attenuation bctwccn any two channcls* 

Intclligiblc crosstalk 65 dB 

Unintelligible crosstalk 58dB 

Maximum idle channcls noisc with all channcls loadad 

cxccpt thc one measured. rrtcrcncrd to peak signal 

point * 

- 8OdB 

Maximum in - band r ms nonlinear distortion, rrtcrcncrd 

to peak signal point * -40dB 

Out -of - band signalling at 3825Hz 

Ptlot frtqumcics at 3 920Hz 

From the theory of analog signal processing, three basic 

methods are known for producing a single-sideband signal. 

These methods are described at length in an overview paper of 

Kurth [ 121, for which reason the principles will only briefly 

be touched upon here. All three methods use for frequency 

translation a type of doublesideband amplitude modulation. 

The differences lie in the sequential order of modulation and 

sideband suppression, as well as in the method of sideband 

suppression itself. Mathematically, the single-sideband signal 

may be expressed by means of analytical signals 

y,(t) = x(t) cos 2rfCt - 2(t) sin 2nfct 

where 

y,(t) = x(t) cos 2n fc t + 20) sin 2nfc t 

(la) 

*Measurement of the multiplexed analog signal, the digital ports being 

looped. 

are synchronized, and the particular signal levels are adjusted 

so as to enable a subsequent digital processing. This is followed, 

for each of the branches, by a serial-parallel (SIP) converter, 

which converts the interleaved TDM signals to parallel 

form. The samples, in Europe encoded according to the A-law 

[10], are converted in the expander (EX) to linearly encoded 

samples. Subsequently, the FDM signal is digitally generated 

by means of a bank of single-sideband modulators (SSB-MOD). 

A digital-to-analog (D/A) converter, followed by an analog 

filter for smoothing the resulting staircase function, produces 

the desired FDM signal at the supergroup level. 

If we consider the operations of Fig. l (a) in reverse sequence, 

we obtain the FDM-TDM conversion shown in Fig. l(b). In 

general, the frequency- to time-division-multiplex conversion 

may be derived from the TDM to FDM conversion by transposition 

[ 1 1 l. For this reason, we can limit ourselves in the 

following to the various methods of producing singlesideband 

signals; thus, only the subassembly SSB-MOD will be treated 

in detail. Suffice to say that a North American or Japanese 

24-channel PCM TDM to FDM group-band translator (with a 

bit rate of 1.544 Mbit/s and inherent p-law companding [10]) 

is made up of the same building blocks as a 60channel transmultiplexer 

according to Fig. 1. 

The remaining questions to be considered in this section are 

primarily concerned with allowed signal irnpairnients-noise, 

crosstalk, phase, and frequency distortion, and nonlinear distortion. 

The requirements common to all transmultiplexing 

represents the Hilbert transform of the input signal x(t) and 

fc the carrier frequency. Equation (la) denotes the upper 

sideband and (lb) the lower one. 

A. Hartley S Method [13] 

The immediate implementation of (1) leads to the Hartley's 

method. The input signal is modulated onto the carrier 

cos 2rfct and the Hilbert transform of the input signal onto 

the orthogonal camer sin 21Tfct. After that, the two branch 

signals are subtracted from or added to each other, depending 

on whether the translated spectrum so produced is to be noninverted 

or inverted. 

Fig. 2(a) depicts the principle of implementation of this 

method and Fig. 2(b) the associated frequency spectra, from 

which the operating principle becomes clear. The unwanted 

sideband is cancelled by compensation. 

A practical implementation of this method is shown in Fig. 3. 

In this case, two signals, shifted in phase through all-pass networks 

by n/2 with respect to each other, are derived from the 

input signal. 

B. Weover S Method (2 41 

The baseband signal is first translated by means of an auxiliary 

carrier (Fig. 4(a)), whose frequency normally Lies in the 

center of the usable band (Fig. 4(b)). Figs. 4(c) and (d) 

depict the corresponding frequency spectra for in phase and 

quadrature signals upon this translation. The low-pass filters 

that follow (LP in Figs. 4(a), (e) suppress the spectral compo-


I cos ZTtf,t 

(b) 

Fig. 2. Singlesideband modulation according to Hartley's method. 

f 

cos 2KfCt 

l 

sin ZTCfCt 

Fig. 3. Modified SSB-MOD according to Hartley. 

nents for f > If0 I (Figs. 4(f), (g)). This is subsequently followed 

in each branch by a translation into the ultimate frequency 

band (Figs. 4(h), (i)). Through addition of the two 

signals X gi(t) and X 3q (t) (Fig. 4(a)), the frequency components 

within the useable frequency band are compensated in 

such a way that the desired single-sideband signal is produced 

(Fig. 46)). 

C. Filtering Method 

The oldest and in analog signal processing normally used 

method is the double-sideband amplitude modulation with 

subsequent suppression of the unwanted sideband (Fig. 5). As 

filter, either a low-pass or a high-pass filter may basically be 

used, depending on whether the noninverted or inverted signal 

Fig. 4. Single-sideband modulation according to Weaver's method. 

spectrum is desired. In general, however, modulation products 

of nonideal modulators have to be filtered out, so that a bandpass 

filter must be used. 

IV. DIGITAL SINGLE-SIDEBAND MODULATION 

The principles of singlesideband modulation outlined for 

analog systems in the preceding section can also be used with 

digital signal processing. The frequency spectra are now 

periodic with the sampling frequency fAY (Fig. 6(a)). 

For estimation of the total amount of digital circuitry, the 

following considerations concerning implementation are made.

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTZPLEXING METHODS 1423 

Fig. 7. Digital FDM signal. 

The amount of circuitry, particularly the number of arithmetic 

operations, that would arise if such a transmultiplexer were to 

be implemented, is so overwhelming that any further consideration 

is superfluous [ 151. 

If efforts to process the digital signals at lower sampling rates 

should prove to be successful, drastic savings could be achieved 

for the following reasons: 

2 Lowpass attenuation 1 

Highpass attenuation 

X (11 : Premodulation 1 baseband spectrum 1 

YE If) : Two sidcband amplitude modulation 

Yu(f): 

Single sideband spectrum (noninverted position1 

Y, If): Single sideband spectrum {inverted position) 

Fig. 5. Single-sideband modulation according to filtering method. 

1) the amount of arithmetic operationsltime is smaller; 

2) the degree of the necessary filters is reduced since the 

relative width of the transition band is expanded. 

If these considerations are pursued consequently, the minimum 

expenditure will be reached if each channel signal is 

processed just with the lowest possible sampling frequency according 

to the Sampling Theorem [ 161. In order to be able to 

produce the FDM signal, the sampling frequency must subsequently 

be increased, requiring special interpolators. These 

interpolating networks wiU be treated in more detail in the 

next section. 

If the sampling rate of a sequence is to be increased, additional 

sampling values must be inserted. If the sampling rate is 

to be increased by an integer factor M, the new sampling interval 

becomes 

is = 

(c) 

Fig. 6. Digital singlesideband modulation. 

= sampling frequency 

If it is desired to form the FDM signal (Fig. 7) of, say, a 60- 

channel transmultiplexer, a bank of 60 singlesideband modulators 

would be required. If the signal bandwidth of a single 

channel is assumed to be BK = 4 kHz, the FDM signal needs a 

bandwidth of (cf., Fig. 7) 

To be able to perform analog filtering after the D/A converter 

(Fig. l), an unconstrained transition band has to be provided. 

Approximately fSp = 16 kHz are assumed for this band, yielding 

an analog filter of tractable order. Since, according to the 

Sampling Theorem the sampling frequency has to be at least 

double the maximum signal frequency, the sampling frequency 

is (cf., Fig. 7) 

where To is the reciprocal of the original sampling frequency. 

Each Mth sample of the new sequence is taken over from the 

sequence to be interpolated; M - 1 new samples are inserted 

inbetween. Two methods for solving the interpolation problem 

are outlined in the following (a more detailed treatment 

can be found in [92]). 

A. Interpolation by Low-Pass Filtering 

The time-continuous signal ?(t) is assumed to have the 

frequency spectrum X(w) (Fig. 8(a)), which shall be band 

lifited according to 

2(0)=0, for 101 > ug. 

The signal i(t) is sampled with the sampling frequency 

fSo = l/To assuming the Sampling Theorem to be met. Due to 

the sampling process the spectrum of i(t) is periodically repeated 

on the frequency axis (Fig. 8(b)), yielding 

Starting with the sequence {~(VT~)} the sequence 

(xH(vTo/~)) is to be approximated by interpolation. This 

sequence could be obtained by sampling the time-continuous 

signal x(t) with sampling frequency fSH = M/To increased by


Fig. 8. Frequency spectra resulting from increasing the sampling 

frequency by a factor of M. 

* 

0 T 2T t 

y(VT)* x(VMT-iT) M c HI 

150 

Fig. 9. Generation of the sequence {JJ(VT)} from .M sequences with 

sampling interval MT (example M = 8). 

Fig. 10. Polyphase network for sampling rate increase. 

the factor M compared with the sampling process of (3). As 

corresponding spectrum we get 

M +- 

In accordance with the increased sampling frequency the 

period of the frequency spectrum is increased by a factor of 

M (Fig. 8(d)). Digital interpolation can be interpreted as a 

low-pass filtering process where the signal is to be amplified by 

XH(u) = - C X(W + MiuSO). (4) a factor of M (Fig. 8(c)). The unwanted spectral components 

To i=-- 

(cf., Figs. 8(b) and (d)) are suppressed; in this way, the spec-

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTIPLEXING METHODS 

-,j 

-815 G 

-l 

-r 

L---- 

I------, 

I----- @ 

-wn 1 

.-.-. 

L.. -. .... ., 

L ----. _ 4 

I : . . ; r -+ 

1 

Fig. 11. Phase cancellation process in a polyphase network for interpolation 

(M = 5). 

trum XH(w) is approximated. An exact and detailed description 

of this process can be found in [17], [181, [92]. 

B. Interpolation by Phase Compensation 

For a simple explanation of the phase cancellation process 

M - l sequences {X~(VMT)) are derived from the sequence 

{X~(VMT)} with the sampling interval To = MT in such a way 

that the sequences result from each other by successive phase 

shifts corresponding to the time interval T (Fig. 9). 

All these subsequences are interleaved to produce the interpolated 

output sequence ~(vT)), sampled at the M times 

higher rate M/To = l/T. The corresponding network structure 

is shown in Fig. 10, where the summing point represents the 

interleaving process. As mentioned in the preceding section, 

the desired interpolation values to be inserted can only be 

produced if the polyphase network as a whole (Fig. 10) 

exhibits a low-pass characteristic corresponding to Fig. 8(c), 

[ 191, [ 201. From Fig. 9 it is obvious that the signal sequences 

in the individual branches of the polyphase network (Fig. 10) 

can be processed at the lower input sampling rate l/MT= 

l . Furthermore, it will be shown that any rational 

(low-pass) transfer function for decimation or interpolation 

can be processed at the lower input sampling rate l/(MT) = 

tation according to Fig. 10. 

If each branch filter is provided with the all-pass characteristic 

H? = eioi shown in Fig. 11, the following output spectrum 

is obtained (a = 27~Tf): 

Range 

(b) 

Fig. 12. (a) Decomposition of branch allpass networks. (b) Polyphase 

structure for interpolation. 

since 

j(anp)i = (summation orthogonality of 

2 trigonometric functions) 

i=O 

This represents ideal low-pass filtering according to Fig. 8. 

The representation of Fig. 11 permits a different derivation of 

the low rate signal processing in the polyphase network: The 

phase characteristics of the branch filters can be decomposed 

in a linear component Gli and in a component periodic 

with 1/MT (Fig. 12(a)). The linear component can be 

realized by delays; solely the periodic component has to be 

realized by all-pass networks H ~(z~) (Fig. 12(b)). Note that a 

reference filter in the zeroth branch of the polyphase network 

(Fig. 12(b)), a prerequisite for realizability, is omitted for 

convenience. 

C. Digital Signal Interpolation in Conjunction 

With Filtering [l 91 

Each sequence f~(v)) whose unilateral z-transform

1426 PROCEEDINGS OF THE IEEE, VOL. 69, NO. 11, NOVEMBER 1981 

exists, can be expressed by N interleaving sequences 

In it, the expression 

N-l 

Y(z) = C z-" y(zN, n). (6 

n =O 

where 

and 

h (0) = lim H(z) 

Z-+ m 

(1 lb) 

R, = h (z - z,,)[H(z) - h(O)]. (I lc) 

m 

Z'zmm 

y(zN, n) = 2 ~ (VN + n) z - ~ (7) For the derivation of the branch filters H,(z~) from the polyv=o 

nomial form according to (9), a method can be found in [l 91. 

represents the z-transform of the subsequences. This process is However, a general closed form solution cannot be given. For 

visualized by writing the terms of the sequence as follows: this reason, the form c according to (l l) is used in the follow- 

~(vT)) = ~ (0) 

Y (NI 

Y(2W 

Y (3N) 

N sibseqgences 

with sampling ~(vN)) 

interval N T 

z-transforms of 

subsequences 

y(zN> 

This results in the following. The fdtering process of a se- ing. Here, one obt@ns after a tedious but straightforward 

quence with the sampling interval T can be reduced to fdtering calculation 

of N sequences having a sampling interval of N - T (Fig. 13). 

Evidently, the transfer function of the polyphase network is M R, 2;l 

H, (zN) = h(0) + C 

(12) 

(Fig. 13) 

m = I P - 2 m 

Starting with (g), the question arises how to get the transfer 

functions of the individual branch filters Hn(zN) in order to 

realize a given transfer function H(z). For this purpose H(z) 

may be expressed in one of the following three equivalent 

forms which can, if necessary, be transformed from each other 

l211 

a) Polynomial Form 

uk' 

k=O 

H(z)= where K

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTIPLEXMG METHODS 1427 

For better understanding of this method, we set out from 

the model that each branch with the bandpass filter transfer 

function H,(z) with complex coefficients3 (Figs. 14(a), (c), 

(d), (e)) is replaced by an interpolating filter for generation 

of the high sampling rate fS = 1/T, followed by a SSB-MOD. 

Moreover, by means of these interpolating filters analytic 

baseband signals, i.e. signals with suppressed power spectrum 

in the frequency interval (-fS, O), have to be derived from the 

real-valued input sequences {x,(vNT)}. Hence, as outlined in 

the preceding section, the individual bandpass filters may be 

composed of a polyphase network with complex coefficients 

and a SSB-MOD (Fig. 14(h)). Thus the following formula 

applies: 

Fig. 13. Filtering by polyphase network. 

analysis and synthesis [26]-[29], as well as for spectral an&- 

sis [221, [3O], [31]. Clearly, the algorithms then are adapted 

to the specific problems to be solved. Note, however, that we 

Limit ourselves to transmultiplexing applications. To this end, 

in this section, the transmultiplexing algorithms are classified 

into four groups. These are: bandpass filter-bank, low-pass 

filter-bank, Weaver structure, and multistage modulation 

methods. 

A. Bandpass Filter-Bank Method 

The frequency spectrum of' the TDM input sequence 

{x,(uNT)} (cf., Fig. 14(b)) exhibits the following characteristics: 

assuming a complex-valued output sequence G~(T)). Thereby 

the polyphase network realizes the frequency response 

~ ~ ( ein i all ~ branches ) according to Fig. 14(a). The subsequent 

single-sideband modulator causes the corresponding 

frequency translation along the frequency axis. From (8) 

f 0110 ws 

According to the modulation theorem of the z-transform the 

ideal complex single-sideband modulation is represented by 

substitution of the variable 

&(z) is thus obtained from H;(Z) by 

2) Frequency response has Hermitian symmetry, ie., 

~,(~iznTfN) = X,*(e-i2"TfN) (15) 

(* denotes "conjugate complex"). 

Starting with a TDM-sequence according to (14) and (IS) 

the corresponding FDM signal may be gained by suitable filtering 

in a bandpass filter bank (Fig. 14(a)). Such a bank consists 

of N bandpass filters, whereby the passband range of the 

individual bandpass filters is shifted by the frequency l/(NT) 

with respect to each other (Figs. 14(c), (d), (e)). The output 

signals of the bandpass filter bank are combined by an adder, 

thus forming the FDM signal. For all channels with 

r E [N/2, N - l], the resulting spectra are in the inverted 

position. If these signals are subjected beforehand to a doublesideband 

amplitude modulation with fs/2 = 1/(2T) as carrier 

frequency, the desired position of the frequency spectrum 

will be obtained. This modulation process is particularly 

simple, since it can be performed by multiplication with the 

sequence 

The FDM signal then exhibits the correct frequency position 

for all channels. In the sequel, this inversion process will no 

longer be mentioned explicitly. 

2 represents the set of integer numbers. 

According to (16) the real single-sideband signal is obtained 

by 

For better understanding, the complex SSB-MOD is depicted 

symbolically in Fig. 14(i). Using (17) and (18), there follows: 

since e-jZar = 1 because of r €N. 

Since the same transfer function &(z) (Fig. 14(h)) is generated 

in each filter branch r (Fig. 14(a)), the following relation 

holds: 

Inserting equations (19) and (20) into (16), we obtain for the 

3Complex signals and transfer functions with complex-valued coefficients 

are indicated by boldface letters and underlined in the figures.


(€9 

Fig. 14. Bandpass fdter-bank method. 

real-valued output signe 

Interchanging the sequential order of summation and rearranging 

yields 

The term 

represents the inverse discrete Fourier transform (IDFT) for 

N points at the I/(NT) sampling rate. We thus.obtain from 

(21) 

N-l 

Y(z) = C z-" 2 Re L~n(zN) - IDFT { xr(P) )l (22) 

n=O 

from which the block diagram in Fig. 15 can be derived. The 

filters H,(z~) with complex coefficients process complexvalued 

input signals. From the output signal, only the real 

part is needed. 

The transmultiplexers described by Terrell and Rayner [32], 

Maruta and Tomozawa [8], [g], [33], [34] as well as by 

Takahata et al. 1351 work in accordance with the bandpass 

filter-bank method treated above. 

If the number of channels is specifically fixed to


(i) 

Fig. 14. (Continued). 

TDM - signal 

sampling rate 

R, = real part 

I, = imaginary part 

1 

m 

Fig. 15. Block diagram of transmultiplexer according to the bandpass 

filter-bank method (see (22)). 

FDM- signal 

where 

the described transmultiplexer method may be decomposed 

into subunits consisting of transmultiplexer~, each of them 

processing $I input signals. The total number of subunit transmultiplexers 

is then 

N N N N 

+p"-


, 

spectral inversion 

2 point lDFT 

Fig. 16. Transmultiplexer for N = 2 channels according to the bandpass 

ffiter-bank method. 

Using (23), s becomes 

The simplest transmultiplexer is obtained for p = 2. From (21) 

results 

Since in this particular case (N = 2), no imaginary parts occur 

in the discrete Fourier transform, we obtain the block diagram 

shown in Fig. 16. Hence, H,(Z~) and H~ (z2) represent filters 

with real coefficients. 

If N is chosen such that 

the entire transmultiplexer may be constructed from stages 

according to Fig. 16. By combining the processors of all subunits 

to an overall processor, we obtain a "Hadamard processor." 

Within this processor only trivial multiplications with 

plus or minus one have to be carried out. A transmultiplexer 

according to this principle has been proposed by Qaasen and 

Mecklenbrauker [3 6 1. 

Furthermore, there exist other methods mechanizing a bandpass 

filter-bank in the frequency domain. They are based on 

fast convolution techniques applying fast Fourier transforms 

(FFT) [37], [38] or number theoretic transforms (NTT) [39], 

[40]. However, these methods seem to be less suitable due to 

their inherent time delay, caused by block processing of the 

incoming sequences. A transmultiplexing algorithm in which 

the bandpass filter-bank is implemented via fast convolution 

techniques is presented by Constantinides and Valenzuela 

[41]. Sophisticated simplifications were made to reduce the 

computational load, for instance, by taking FIR filters. To 

meet the specifications (60-dB minimum stopband attenuation, 

passband gain accuracy k0.5 dB), which do not comply with 

the CCITT Recommendations according to Table I, a linear 

phase FIR filter degree of about 1760 is needed for a 60- 

channel transmultiplexer. In conjunction with the minimum 

block length this leads to a (looped) group delay of greater 

than 3 ms (cf., Table 11). Moreover, it is conjectured that even 

with the application of minimum phase FIR filters a group 

delay of less than 3 ms cannot be reached. 

Most recently, a modulator-free and multiplier-free transmultiplexer 

approach was proposed by Kurth et al. [81 I, [821. 

It resumes the most obvious idea of directly selecting the 

desired frequency band, for instance of the periodic input 

spectrum of the TDM signal (TDM-to-FDM), at' the group or 

supergroup level by means of a single filter per channel-a 

method which has so far been abandoned due to its extremely 

high computational load (cf., [ 15, ch. 41 ). The per-channel 

processor uses minimum-phase FIR filters which also perform 

interpolation (TDM-to-FDM) and decimation (FDM-to-TDM). 

Hardware efficiency, which can hardly be compared with that 

of other approaches, is obtained by implementing the FIR filters 

in a logarithmic mode, wherein multiplication becomes 

addition. Lookup tables, in form of memoxy, perform conversion 

between the logarithmic and linear formats. 

A similar transmultiplexer with conventional digital filters 

processing complex (analytic) instead of real signals was proposed 

by Del Re [83]. Before feeding the input sequences to 

the complex filter-bank on a per-channel basis, the analytic 

signals are generated by means of a phasesplitting network 

(Hilbert filter). 

B. Low-Pass Filter-Ban k Method 

If one succeeds in obtaining real signals at the output of the 

Fourier processor, processing of "complex" signals in the poly- . 

phase network can be avoided. The output signals of a Fourier 

processor are real if the complex input signals exhibit 

Hermitian symmetry. Assuming a 2Wpoint DFT processor, 

the following relation must hold for the complex input signals: 

For a 2N-point inverse discrete Fourier transform, the following 

holds: 

Using (26), equation (27) may be factored into 

Applying the variable transformation 

to the second term of (28) yields 

Thus there results from (28), if in its first term r is replaced


TABLE I1 

COMPARISON OF THE ~NDIVIDUAL TRANSMULTIPLEXER APPROACHES 

1) GROUP DELAY MEASURED AT THE ANALOG PORTS, THE DIGITAL 

PORTS BEING LOOPED 2) PELLONI'S FILTER BANK (POLYPHASE NETWORK) 

Transmult~plexer 

approach 

Tcrrell W75 [32] 

Maruta 1976 [33] 

lakahata 1978 

[3L,35] 

:loam 1978 [36] 

Aoyama 1980 [ 8,9 1 

Constantinides 1980 [L11 

hdul. scheme, 

40 of chan~ls: 

jingle -way IS) 

'WO-way IT) 

bltiple-way (M1 

M 12 (2LI 

M M) 

M 12 (24 

Group 

delay '1 

? 

2.3ms 

> 3rns ? 

? 

2.3ms 

-3ms 

Operation 

rate lchmtl 

[l/sI 

Degree 

of 

modularity 

IOW 

l ow 

low 

~ d i u m 

low 

low 

Analog 

frequency 

cowersion 

Yes 

M 

no ISG: yes) 

Yes 

no 

Yes 

Rocessing 

of 

out - of -band 

- signalling 

separate ? 

inherent ? 

scparate ? 

- 

inherent 

separate ? 

Bellanqzr l97L [L2,LL,69] 

Tomlinson l976 [L81 

Drogespt R78 [L61 

Rdste [L71 

Vary 1978 [22.30] 

Pelloni 1979 1231 

Narasimha 1979 [SO] 

2.5ms 

c 3ms? 

-3ms 

Not S 


('J) 

Fig. 17. Generation of an FDM signal according to the low-pass filterbank 

method. 

Fig. 18 depicts the block diagram of the transmultiplexer 

operating according to this method. 

A comparison of Fig. 15 with Fig. 18 shows that there are N 

complex bandpass filters required in the bandpass filter-bank 

method and 2N low-pass filters in the low-pass filter-bank 

method. The number of filters is the same, since a complex 

filter can be constructed from two real filters. As the sampling 

rate in the low-pass filter-bank method is by a factor of two 

lower and the absolute widths of the filter transition bands are 

the same, the amount of circuitry for the low-pass filters is 

smaller than that for the bandpass filters. On the other hand, 

the number of arithmetic operations in the Fourier processor 

remains about the same, because a 2N-point IDFT at half the 

sampling rate requires about the same amount of circuitry as 

an N-point IDFT at double sampling frequency. 

A considerable part of the circuitry to be expended on the 

low-pass filter-bank method is required for the generation of 

the complex singlesideband signals. Here, the Hartley method 

proves to be the most favorable, for it needs about 4 poles and 

4 zeros for a phase splitting network. In contrast, the Weaver 

method requires about 8 poles [3]. 

In 1974, Bellanger and Daguet [42] first proposed a transmultiplexer 

according to the above mentioned low-pass filterbank 

method. Further modifications of the DFT-algorithm 

lead to a special inverse discrete Fourier transform (102 DFT). 

Moreover, filters generating complex-valued signals can completely 

be omitted. Thus a further reduction of the number of 

arithmetic operations is accomplished. This highly efficient 

type of transmultiplexer was presented by Bonnerot et al. 

[43] -[45], by Drageset et al. [46], R&te et al. 1471, and by 

Pelloni [23]. 

Still another transmultiplexer, proposed by Tomlinson and 

method. In this approach polyphase filtering is accomplished 

by a single time-multiplexed filter. This filter is mechanized as 

a so called time-varying filter where the coefficients are 

periodically interchanged at a rate corresponding to the low 

sampling frequency. Thus the effective operation rate complies 

with the high sampling rate. At first sight, the transmultiplexer 

of Tondinson and Wong [481 seems to be of lower 

computational complexity than the Bellanger and Daguet [42] 

approach. Yet the computational load of the polyphase filters, 

operating at the low sampling rate, is about the same as that of


TDM-signal 

FDM -signal 

SS0 : single-sideband modulator (Hartley - or Weava- method] 

sanp(ing rate 

V' W 1 

- 1 1 1 

J, I 

NT 2NT T 

Fig. 18. Block diagram of transmultiplexer according to the low-pass 

filter-bank method. 

the time-varying filter operating at the high sampling rate. 

Thereby the necessary control circuitry is not considered. 

Finally in 1979, Narasimha and Peterson [50] presented a 

transmultiplexer using an N-point discrete cosine transform 

(DCT). In contrast to other low-pais filter-bank methods, with 

this approach no premodulation is required to generate complex-valued 

signals. This means that the individual filters of the 

filter bank to be derived from a low-pass prototype filter 

exhibit Hermitian symmetry according to (15) corresponding 

to real coefficients. Thus, in order to obtain the desired passband 

frequency position of the individual bandpass filters of 

the filter bank, the low-pass transfer function has to be 

shifted by 

instead of 

Summation of the individual channel signals yields the output 

signal 

. [H,(~N, in[r+llZl ) + H, (p, -in [r +l12 l )l. (34) 

By inspection and employing trigonometric identities we get 

after interchanging the sequential order of summation 

. cos [d2r + 1) 5 

l 

- x,.(z~). (35) 

r=o 

n 

Here, the second term represents the DCT. 

substitution 

Applying the 

as it is the case with the other low-pass filter-bank methods. findy we obtain 

In conjunction with Hermitian symmetry and the Modulation 

Theorem of z-transform we get N-l N-l 

Y(z) = C z-"G,(z~) cos 

= 3 [ ~ ( ~ ~ 1 i IN) ~ + [ H(ze-jn ~ + ~ lr+llZ / ~ 1 IN)]. (32) n =O r=0 

Comparing this solution with (31), here only an N-point DCT 

~pplyin~ the polyphase principle for interpolation we derive processor instead of a 2N-point DFT processor is needed. The 

with (8) 

branch filters of the polyphase network are modified according 

to (36). With the application of linear phase FIR filters 

significant simplifications of hardware are possible, as it is 

reported in [ 501. A somewhat different implementation of 

the Narasimha and Peterson approach was proposed by 

Marshd [80]. Here the DCT is performed in an other way 

and the filters of the polyphase network are specifically 

matched single-input-double-output filters of the recursive 

(33) type.


H, = rcurs.~ lwrpols 

H2= transversal lowpau 

1 voice -band spectrum 

I , 

flkHz 

('J) 

Fig. 19. Block diagram of the transmultiplexer according to Freeny 

etal. 1151. 

C. Weaver-Structure Method 

A direct method for the generation of an FDM signal is the 

Weaver SSB-MOD bank according to Section 111-B. By ingenious 

partitioning of filters into recursive and transversal lowpass 

filters, operating at different sampling rates, the amount 

of circuitry can be kept low. Fig. 19 shows a transmultiplexer 

according to Freeny et al. [15], [51] which operates in compliance 

with this method. 

The TDM signals are being furnished at a sampling frequency 

of fsl = 8 kHz and modulated onto an auxiliary carrier with a 

frequency of fs1/4, so that the voice signal occupies the frequency 

range -2 < f/kHz < t 2. The modulation is very 

simple, since the sampled carrier is given by the sequence 

{. . - 0, 1, 0, - 1, 0 . . .). In the next step, the sampling fre- 

quency is doubled. The filter HI represents a recursive lowpass 

filter with a cutoff frequency of 2 kHz (Fig. 19(b)). At 

the output of filter HI, the sampling frequency is increased by 

a factor of M. This signal is processed by the following transversal 

low-pass filter H*. After that, the resulting spectra are 

shifted to the desired frequency bands by modulation. The 

FDM signal is obtained by summation of the individual signals. 

The purpose of the two low-pass filters is twofold: They are 

necessary for the operation of the Weaver modulator, and they 

serve at the same time as interpolation filters as described in 

Section V-A. 

Almost one year before the publication of the above Freeny 

papers, Kurth [52], as well, proposed a sophisticated transmultiplexer 

scheme based on the Weaver modulation. While feed- 

ing the Weaver modulator by highly oversampled, ingeniously 

interpolated, and spectray inverted voice-band sequences, he 

uses high-pass instead of low-pass filters in the Weaver modulator. 

However, the main drawback of this proposal is that, in 

contrast to the Freeny approach [l 53, the Weaver modulator 

operates at the high sampling rate throughout. Therefore, the 

resulting operation rate is far too high for a practical implementation 

(cf., Section IV). According to the proposal of 

Kurth an exploratory TDM/FDM single-sideband generator 

was presented by Kao [53]. 

An extension of the approach reported by Freeny et al. [l51 

was-even earlier in time than [52] but not recognized in its 

far-reaching importance in those days-described by Darlington 

[ 161. To thls end, the low-pass filter cascade 

In this expression, [HR (z)]-l represents the recursive part and 

Hr(z) the transversal part of the transfer function H(z). As 

outlined in Section V-C, the transfer function H(z) can be split 

into parallel branches according to (8)


As can be seen from (13), the recursive part can be put as a 

factor in front of the summation, since it is common to all 

additive terms. Thus using (38), equation (39) becomes 

N-l 

H(z) = [ H~(z~)I-' - C Z-'HTr(zN). (40) 

r=O 

According to (40), the transmultiplexer of Fig. 19 is modified 

such that the structure depicted in Fig. 20 results. The modulators 

at the input are now replaced by rotating switches. 

The part of the transmultiplexer in Fig. 20 bounded by the 

dashed Line can be simplified further. To this end, the FDM 

L__-- - --J 

Fig. 20. To the derivation of Darlington's transmultiplexer [ 161. 

Furthermore, the following expressions hold: 

Yini(z)=.fhi(zN) N-l C Z-'H~(~) 

r=o 

N-1 

Yqi(z) = kqi(zN) C z-~H~~(z~). 

r=o 

(44) 

(45) 

Inserting (44) and (45) into (42) and (43), respectively, we 

signal is expressed in terms of symbols used in Fig. 20 Yj&) = ihi(zN) C {z' 3 [e+i(2nlwir 

r=o 

N-l 

According to the Modulation Theorem of z-transform, we ob- 

N-l 

yqi(z) = iqi(p) {z-r -L [e+f(2n~~)ir 

tain for Yini and Yqi r=o 2i 

cos X = + (eix -t e-ix)


1 

sampling f rcqucncy 8 kHz 1 8 N kHz 

Fig. 21. Transmultiplexer according to Darlington [ 161. 

and 

the summed signal at point 1 is 

and by inserting (46) and (47) into (411, there results 

Y(Z) = k*i . z-~ 

i=o { 

N-l 

rq 

cos (: i$ 

HTr(zN) 

An interchange of summation and rearrangement of factors 

leads to 

where fo = fs/4. 

The spectrum at this point is given by 

+Xz(f- fo)+Xz(f +fob 

In the subsequent low-pass filtering, frequencies higher than 

l fo l are suppressed, so that the signal at point 2 becomes 

At point 3, we obtain 

From (44) the transmultiplexer according to Darlington [l 61 

with a cosine and a sine processor can be derived, the block 

diagram of which is shown in Fig. 21. 

Note: If the sampling rate is reduced by a factor of two at 

the output ports of Ihe recursive filters HR(zN), and, in addition, 

the cosine-sine processor is provided with the conjugate 

complex input signals, a W-point DFT results as can easily be 

shown. The transmultiplexer of Fig. 21 then merges into that 

of Fig. 18. 

A modified version of the transmultiplexer of Fig. 21 was 

proposed by Singh et al. [54]. By suitably processing the 

branch signals of the Weaver modulator, they achieve the conversion 

of two TDM signals using only one modulator (Fig. 

22(a)). The sum of the two signals to be processed is applied 

to the upper branch, whereas the difference of these signals is 

fed into the lower branch. 

According to the Modulation Theorem, the z-transform of 

Ydf)=4[~1(f-fo -fc)+X1(f-fo +fc) 

+Xz(f-fo -fc)+Xz(f-fo +fc)l. (50) 

Using similar calculations, the signal at point 3' of the quadrature 

branch is obtained 

Y3r(z) = - 3xl (ze -jzrrfalfs 

-i~nfclfs) 

+ $xl (ze -iznfO Ifs .e +iznfclfs) 

+ ~ .~~(~~-i2nf~ Ifs .e-iznfc/fs) 

- (,,-iZnfo Ifs +iZnfclfs 

2 2 'e 1

(b) 

Fig. 22. 'Modification of the Weaver modulato: according to Singh 

et al. [54]. 

Using (50) and (51), there results at the output 

This process is depicted in Fig. 22(b). Now only N/2 recursive 

low-pass filters are needed instead of N recursive low-pass 

filters. The same applies to the transversal filter bank at the 

output (Fig. 21). The N-point sine and cosine processor is 

reduced to one with N/2 points. However, processing of the 

signals must now be done at double speed, because the sampling 

frequency is increased by a factor of 2 to 16 kHz. Therefore 

a reduction of the necessary amount of circuitry and 

arithmetic operations cannot be expected. 

Similar signal processing is also possible using a HartIey 

modulator. With this method, two channel signals may likewise 

be converted using one modulator only [4]. 

Recently, an alternative approach to the Weaverstructure 

method was published by Peled and Winograd [55]. The 

complex input signals are generated by Weaver modulators. 

In contrast to the transmultiplexers described so far in this 

section, linear-phase transversal filters are used exclusively 

both for band limiting and interpolation. By suitable grouping 

and utilization of symmetries, a considerable reduction of the 

number of arithmetic operations is achieved. The resulting 

signal processor is specifically tailored for a TDM/FDM conversion 

of 12 channels. To realize a 60-channel transmultiplexer, 

five such transmultiplexers have to be connected by 

using an analog interface. 

D. Multistage Modulation Methods 

With muitistage modulation methods, both the sampling 

rate and the spectral position of the signals to be processed are 

changed step by step. Various intermediate FDM signals are 

composed which comprise an increasing number of channels 

from stage to stage. In the last stage the FDMsupergroup 

signal is formed. 

A transmultiplexer operating according to this principle was 

published by Tsuda et al. 1561, [58]. The principle is explained 

by way of an eight-channel transmultiplexer shown in 

Fig. 23(a). For convenience, complex signals are used throughout; 

all spectra-apart from those of the real input sequences 

{xi(v8T) l i = 0, 1, 2, - .., 7)-are spectra of complex sequences. 

The symbols used for multiplication, addition and 

interpolation of complex sequences in Fig. 23(a) are explained 

in the time domain in Fig. 23(b). The spectra depicted in Fig. 

24 serve as illustration for the signal processing in the trmsmultiplexer; 

the notation used refers to Fig. 23(a). 

The TDM signals Xi(z) (Fig. 24(a)) are fed in at a sampling 

rate of fsl = fs/8 =-8 kHz and modulated on a carrier at a 

frequency of fsj /(4) (Fig. 24(b)). This is the same process 

as in the Weaver-structure method described in Section VI-C. 

After the subsequent low-pass filtering with the filter characteristic 

Ho(eln) (Fig. 24(c)) the spectra Xi2, i = 0, 1, 2, . - - ,7, 

as shown in Fig. 24(d), result. By inserting a zero between 

every two successive samples, the sampling frequency is increased 

by a factor of two, whereby, as outlined Section 

V-A, Ell serves as interpolation filter pair (Fig. 24(e), (f)). 

Thereafter, by a modulation process, the spectrum of every 

even-numbered branch is shifted to the left by Af = 1/(32T), 

and that of every odd-numbered branch to the right by 

Af = 1/(32T). Fig. 24(g) shows the spectra Xi4 for i = 0, 

2,4, 6, and Fig. 24(h) those for i = l, 3, 5, 7. Finally, the 

first stage FDM signals Xis/(i = 0, 1, 2, 3) are obtained by 

addition of the signals Xi4 and X(i+1)4 for i = 0, 2, 4, 6 

(Fig. 24(i)). 

In the following stage the signals are processed in the same 

way (increase of the sampling frequency by a factor of two, 

"left7'- and "right"-shifts of the spectra by Af = 1/(16T)).


traquancy danain 

B(zl {RcCa~t} & 

time domain 

{RC~~IVII} 

RcIc 1~11) 

U 

U 

(b) 

Fig. 23. Eight-channel transmultiplexer according to Tsuda et al. [56].

SCHEUERMANN AND GoCKLER: DIGITAL TRANSMULTIPLEXING METHODS 

Fig. 24. Spectra of transmultiplexer according to Fig. 23. 

The corresponding spectra are sketched in Figs. 246), (k), (l), 

(m), (n). In the last stage the samphg frequency is increased 

once more by a factor of two, however, the modulation 

process is chosen such that the spectrum of the branch i = 0 is 

shifted along the frequency axis .by Af= 1/(8T), and the 

spectrum of the branch i = 1 by Af = 3/(8T) to the right (Figs. 

24(0), (p), (q), (I), (S)). The complex FDM signal Y is produced 

by addition of Xolo and Xllo (Fig. 24(t)). Its real part 

represents the desired FDM signal Y (Fig. 24(u)). 

A great advantage of this method is that it only requires a 

small number of different filter types. go represents a pair 

of identical low-pass filters with a narrow transition band 

(Fig. 24(c)). Because these filters operate at the lowest possible 

sampling rate, the amount of circuitry, however, remains 

relatively low. Furthermore, it can be shown for i > 0 (i Em 

that all filters _Hi exhibit the same relative width of the transition 

band. Thus for every &(i > 0) just one filter type can 

be used; the frequency response is selected merely by changing


R = ZR -f- 

fs 

fs = output sampling rate 


Fig. 25. Transrnultiplexer subunit for 12 charineis according to Fettweis [59]. 

the sampling frequency. In addition, the expenditure for the 

interpolation filters is reduced to a great extent, since nonrecursive 

"half-band" filters [57! are used. In [58] an efficient 

experimental realization of such a 60-channel transmultiplexer 

isdescribed in detail, and experimental results are given. 

Recently, a further transmultiplexer design based on the 

multistage principle of Tsuda et al. was published by Constantinides 

and Valenzuela [73]. Here, for any two intermediate 

FDM-signals complex modulation, interpolation and 

filtering are performed by using only one type of a two branch 

polyphase network, each combined with four operations for 

spectral inversion, and a band multiplexer using an FIR filter. 

h alternative transmultiplexer approach, based likewise on 

the multistage modulation principle, is proposed by Fettweis 

[59]. By appropriate selection of the basic sampling rate and 

by ingenious arrangement of the various carrier frequencies, it 

is accomplished that every modulation process is reduced to a 

multiplication with a sequence of carrier samples only comprising 

the trivial values of 0 or + 1. 

As a consequence, the modulation processes cannot produce

Fig. 26. Spectral illustration of the transmultiplexer subunit according to Fig. 25. 

quantization errors. The principles of such "multiplier-free" 

modulation systems are described in detail in 1601. "Mdtiplierfree" 

modulation schemes for TDM/FDM conversion are outlined 

in [59]. This modulation scheme is particularly favorable 

for the conversion of two 2048 Mbit/s PCM TDM bit 

streams (PCM30) to a supergroup of the FDM hierarchy (60 

channels in the frequency band from 312 kHz to 552 kHz), 

since with this approach an additional analog frequency 

translation of the FDM signal can be avoided. The principle of 

the Fettweis approach will therefore be treated for a 60- 

channel transmdtiplexer. 

The 60 TDM voice signals are converted to the FDM format 

using five submits, each of them for 12 voice channels (Fig. 

25). Subsequently, these five intermediate FDM signals are 

combined to a 60-channel FDM supergroup signal (Fig. 27). 

The spectra in Figs. 26 and 28 serve as illustration; the notation 

of Fig. 25 is related to Fig. 26 and the notation of Fig. 27 

corresponds to Fig. 28. 

The TDM input signals Xb(z) are supplied at a sampling 

frequency of fso = fs/72 = 8 kHz (Fig. 26(ad). BY inserting 

two zero-valued samples between every pair of adjacent 

samples of the original sequence and subsequent filtering with 

the frequency response H. 

(Fig. 26(a2)), the sampling fre- 

quency is increased by a factor of three to fS1 = fS/24 = 24 

kHz. For details on the selection of fs, = 24 kHz, the reader 

is referred to the discussion in 1601. 

By multiplication of the signals of every second channel by 

the sequence ((-l)'), the spectrally inverted signals xAK are 

produced for n = l, 3, 5, . - - , 11 (Fig. 26(a4)); another increase 

of the sampling frequency by a factor of two to



fsz = fS/12 =48 kHz follows (Fig. 26(a5)), Fig. 26(a6) fox 

n = 0, 2,4, . . . , 10 and Fig. 26(a7) for n = 1, 3,5,. - - , l l). 

Furthermore, in the Figs. 26 (a,) and 26 (a7), the carrier fre- 

quency fC1 = fs& is shown. After the modulation process, 

performed by multiplication with the sequence {ol(v. 12~)) = 

the spectra X: are obtained (Fig. 

{. . - 1, 0, 1, - 1, 0, . - m), 

26(a8) for n = 0, 2, 4, .- - , 10 and Fig. 26(a9) for n = 

1, 3, 5, ..., 11). By low-pass filtering (Fig. 26(alo) the upper 

sideband is suppressed (Fig. 26(all) for n = 0, 2, 4, ..., 10 

and Fig. 26(a13) for n = 1, 3, 5, - . - , 11). After the spectral 

inversion of the signals in the channels n = 0, 2,4, ..., 10, the 

signals xK and are added, yielding the signals X; for 

n = 0, 1,2, . - . , 5 (Fig. 26(al4). 

Subsequently the signals X: for n = 1, 2, 3,4 are processed 

in the same way as the signals X;, with the only difference 

that the sampling rate is twice as high at any point (Fig. 26(b 1) 

through (bll). The spectra for X: and X; are depicted in 

Fig. 26(b12). The same holds for the signal (Fig. 26(cl) 

through (cl2) in the next hierarchy stage. 

As shown in Figs. 26(dl) through (dl;), the spectra X; and 

X: are processed differently; the filter characteristic Hg appears 

only in the channels 0, 1, and 10, 11. Finally, the Figs. 

26(e1) through (e4) show, how the signal Yi is generated from 

the signals xk3, xi3, and X:$. 

The subsequent processing of the subunit signals Yi(i= 

0, 1,2,3,4) is shown in Figs. 27 and 28. After having increased 

the sampling rate by a factor of three, the signal Y: 

already occupies the desired frequency band (Fig. 28(al)-(~3)). 

The same holds for the signal Y: (Fig. 28(dl)-(d3)) if it is 

spectrally inverted beforehand. 

In processing the signal Y4 two spectral inversions have to be 

performed (Fig. 28 (bl) through (ba)), one before and one after 

the sampling rate increase. Each of the signals Y1 and Y3 is 

subjected to a modulation (multiplication by the sequence 

(P(v - T)) = 1. - . , - 1, 0, 1, 0, - 1, 0, . . .)) and subsequently 

passed through a high-pass filter for sideband suppression (Fig. 

28(cl) through (~4) and Fig. 28(el) through (e~). Finally, 

the 60channel FDM supergroup signal Y(z) is formed by 

adding the five signals Y;, Y:, Y:, Y& and ~ 4 (Fig. 1 28(f)). 

~ 

The transmultiplexer approach by Fettweis is well suited for 

the application of wave digital filters [61], [78]. A detailed 

discussion on this subject can be found in [591. At present, 

this approach is the only transmultiplexer which might possibly 

be mechanized by means of sampled-data filters (e.g.,

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTPLEXING METHODS 

nbn, AAM, nnnn, AAAA 

o 16 32 LB 6i 80 96 112 128 IU 160 176 192 f 

T 

T 

I 

T 

T 

nnnn 

ILL 160 176 192 

f 

- 

192 f 

AA AA, - 

160 176 192f 

I 

I nnnr\, a 

0 16 32 6L % 128 1M3 176 l92 f 

AA r\nnn,~~nn, nnnn - 

0 32 L8 80 96 112 1U 160 192 f 

nn M, nmnn~, AAAA, - 

0 32 L8 80 96 112 1U 160 192 f 

- 

9 

0 96 192 f 

AAAA 

nnnn, 

0 32 L& 96 lk 160 192 f 

nnnh AAAA, - 

0 L8 6L 96 128 ILL 192 f 

nnnn, AAAA, - 

32 L 8 96 1 U 160 192 f 

nnnnnnnn, 

AAMAAAA, 

0 32 6L 96 128 160 192 f 


switched-capacitor filters 1601, [62]-1641, 1791, because it 

does not make use of cancellation. 

In the previous four sections of this chapter we have discussed 

transmultiplexer methods which are based on one of 

two fundamental concepts of filter decomposition: 1) parallel 

decomposition using a signal processor and 2) serial decomposition 

resulting in a tree-like structure. Naturally, it is 

possible to compound both principles. Recently such an 

approach was disclosed by Molo [74]. This approach starts 

from a 2-kHz frequency shifted version of the incoming 

TDM-signal xr(zN) as it is shown in Fig. 17(a). Now the 

remaining transmultiplexer has to perform a filter bank which 

is derived from a low-pass prototype. The basic idea is now to 

decompose the filter both in a parallel and in a serial manner. 

The manipulation of the parallel part of the filter decomposition 

equals the low-pass principle. The treatment of the serial 

part resembles the derivation of a radix-2 butterfly in an FFT. 

Thus we get a complex 8-point IDFT followed by a tree-like 

structure with multipliers and modulators to perform frequency 

pre- and post-shifting. Note that the number of TDM 

channels to be frequency multiplexed in this approach has to 

be a power of two. 

VII. COMPARING THE TRANSMULTIPLEXER METHODS 

If we want to compare the individual transmultiplexer 

methods described so far with each other, we need an appropriate 

tool for comparison. Certainly, the most important 

features of such a tool are cost and performance. To allow a 

more detailed judgment, however, a representative list of 

criteria will be set up as a basis for the subsequent comparison. 

A. List of Criteria [I] 

When comparing digital transmultiplexing methods with 

emphasis laid on the inherent algorithms, the subsequent list of 

criteria must, to some extent, be incomplete, since not all 

aspects of hardware are covered in this paper. Note that the 

sequential order of the following criteria is chosen essentially 

according to their relative importance in these authors' 

opinion. 

1) 'Stability Under Looped Conditions (6.51: With the application 

of digital transmultiplexers, stability problems may 

potentially arise, since these assemblies are installed in the 

four-wire branch of an essentially two-wire transmission line 

(Fig. 29). Due to the four-wireltwo-wire transitions (hybrids),


* 

0 96 192 f 

x;k'*)l I I I I 

I 

nn, I AA, , ,m I AA, 

0 b 16 

k6,lP)l 

A A, 

32 6L 80 88 96 10L li2 128 160 176 18L 192 f 

I 

I 

I 

\h, , ,AA, I 

, 

I 

- 

0 8 16 32 6L 80 88 96 1OL 112 128 160 176 18L 192 f 

1x;i 

AA, AA AA, M, 

0 16 2L L0 L8 56 72 80 96 112 120 136 1U 152 168 176 192 f 

P ,m nnM, AA, M, AA,~ 

I 

I 

n, nn -m 

0 16 2L L0 LB 56 72 80 96 112 120 136 1U 152 l68 ' lj6 192 f 

lH2l 

IIX:I 

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0 l6 2L . L8 

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L 


transmultiplexers actually operate in a looped arrangement. osciUations will generally disturb not only the telephone 

Stability, however, must be guaranteed under any adverse channel under consideration but also a multiplicity, if not all, 

condition, such as strongly unbalanced hybrids which may adjacent channels. 

occur in practice. This is particularly crucial, since parasitic One way to avoid these parasitic oscillations under any


Fig. 27. Complete 60-channel transmultiplexer according to Fettweis 

[591. 

practical circumstances is to use transversal (FIR) filters. Alternatively, 

wave digital filters (WDF) may be used when the 

exclusive utilization of FIR filters is inefficient. For these 

filter classes, potential stability under looped conditions has 

been proved by Meerkotter and Fettweis [65]. They showed 

that only minor m'easures are required for guaranteeing 

absolute stability. Note that their proof is based on the implicit 

assumption of a single-way modulation scheme, meaning 

that each input signal to the TDM/FDM-translator arrives at 

the output by a separate way. Furthermore, this stability 

theory can be extended to two-way modulation schemes using 

Hartley or Weaver modulators [l 21-[ 141 for processing 

complex-valued signals. However, in order to assure absolute 

stability in this case, the required computational complexity is 

substantially higher than that of single-way modulation 

schemes. Initial attempts to apply this theory to transmultiplexing 

methods utilizing discrete Fourier-, sine-, cosine-, or 

Hadamard-processors, respectively, indicate that, as compared 

to a two-way modulation scheme, still much additional circuitry 

would be required for guaranteeing absolute stability (if 

it could be achieved at all) under looped conditions [ 11. 

2) Absolute Value of the Group Delay: As stated in 

Table I, the absolute value of the group delay measured at the 

analog ports of a transmultiplexer, the digital ports being 

looped, may not exceed 3 ms. This criterion is placed just 

behind stability, since some, even promising transmultiplexer 

approaches do not meet this requirement. This holds true 

particularly for some methods applying linear-phase FIR 

filters, or block processing, respectively. 

3) Computational and Control Complexity: The overall 

computational complexity of a digital signal processing algorithm 

can be measured by the number of multiplies and/or 

adds to be carried out per second. However, these figures are 

suitable measures only if the associated wordlengths are taken 

into account appropriately. This is, for instance, achieved if 

the multiplications and additions are counted at the bit level 

[ 151. mus transmultiplexers which make explicit use of 

multipliers can be compared with other approaches being 

based on the application of the canonical signed-digit (CSD) 

code [Ss], [66]. With this code all multiplications are com- 

pletely replaced by combined shift/add operations. In contrast 

to these ideas, when using higher integrated signal processors, 

the computational load may be measured in machine cycles 

per second [ 21. 

Besides the multiplication and/or addition rate the overall 

complexity of the controlunits (timing, logic, etc.) may not be 

overlooked. Unfortunately, there does not exist any general 

direct relation between the efficiency of a transmultiplexing 

method and its control complexity. It is even felt that the 

more efficient the algorithm the higher the control complexity. 

Hence, concerning the latter, the subsequent comparison must 

compulsorily be rather vague. 

4) Modularity: A highly modular transmultiplexer implementation 

is thought of being composed of a great number of 

identical subunits of low complexity. Moreover, the number 

of different subunits should be as small as possible. For this 

reason, a highly modular transmultiplexer approach has the 

potential of good testability, high reliability, and may, in addition, 

be suitable for LSI/VLSI-integration. Furthermore, the 

overall control complexity is expected to be noticeably smaller, 

since identical subunits can, in general, be controlled by identical 

pulse trains. Finally, design and manufacturing will greatly 

be simplified in contrast to an approach of lower modularity. 

5) Intelligible Crosstalk: In a transmultiplexer residual 

crosstalk between arbitrary channels may occur within the 

limits given in Table I. Clearly, intelligible crosstalk represents 

a more severe system degradation than unintelligible crosstalk. 

Therefore, different transmultiplexer approaches may also be 

compared with each other on the basis of their relative amount 

of intelligible and unintelligible crosstalk introduced. According 

to Fettweis [I] and own investigations, intelligible crosstalk 

can essentially occur only between neighbouring channels, 

a single-way modulation scheme being provided. On the contrary, 

in a transmultiplexer based on a multiple-way modulation 

scheme in combination with a phase-compensation 

method for filtering, in at least every second pair of any two 

channels intelligible crosstalk is observed. 

6) Analog Frequency Conversion: Considering the European 

60-channel transmultiplexer approach, there are two suitable 

rates to be used for sampling the FDM supergroup signal,


~YLK (ejln)l 

A [1;2'\- h 4 L 

0 2L L2 1M 158 216 23 288 312 360 LOB L% 504 552 576 

Wt 

, r-f 

0 2L R 288 SOL 552 576 

Fig. 28. Spectral illustration of the transmultiplexer according to Fig. 27. 

512 or 576 kHz, respectively. The first one (generally leading 

to slightly more efficient digital transmultiplexing algorithms) 

requires an analog frequency translation of the FDM output 

signal to the supergroup band (ranging from 3 12 to 552 kHz) 

after D/A conversion. Thus additional noise is introduced by 

the supplementary analog circuitry [67]. On the contrary, 

with a sampling rate of 576 kHz this drawback can be overcome, 

since the FDM supergroup signal is obtained directly 

in the desired frequency band. Hence, a bandpass filter for 

post D/A conversion smoothing is all that is required. Moreover, 

it can be shown that this bandpass filter exhibits better 

sensitivity properties than the filters needed for the analog 

frequency translation with the 5 12-kHz super-group sampling 

rate [851. 

7) Out-of-Band Signaling and Pilots: Particularly related 

to European transmultiplexer approaches using out-of-band 

signaling, it is essential to know whether the signaling information 

and the pilot tones can be processed in the main unit,

, I ] ,l,2 12 f 

0 2L R 120 163 216 &L 288 312 3&l L08 L 56 50L 552 576 

Fig. 2 8. (Continued). 

under looped conditions has not yet been proved [l]. Some 

- ------ other proposals use a two-way (T) modulation scheme. Fundamentally, 

the required absolute stability can be guaranteed 

Hybrid 

Hybrid 

\ 

B. Comparison of the Individual Transmultiplexer Approaches 

In this section, the attempt is made to compare the transmultiplexer 

methods described so far on the basis of the above 

list of criteria. All available data are compiled in Table 11. It 

must be noted, however, that some figures are more or less 

uncertain as indicated. Moreover, other data are not directly 

comparable with each other, since the prescribed specifications 

do, in many cases, not comply with the CCITT Recommendations 

G. 791 to G. 793, published in 1980. This holds particularly 

true for those designs of earlier days. The sequential 

order of the columns of Table I1 is chosen according to the 

list of criteria of the preceding section. 

Besides the number of channels, it can be seen from the 

second column of Table I1 that the majority of transmultiplexer 

approaches is based on a multiple-way (M) modulation 

scheme. As stated above, 

for these approaches at the expense of considerable additional 

expenditure, the sole application of nonrecursive (FIR) and/or 

wave digital (WD) filters provided [ 1 ], [61], [65]. However, 

such an attempt has never been reported by Kurth [52] and 

+ 

rc-tc 

Kao [53], Freeny et al. [Is], 1511, Singh et al. [54], Tsuda 

et al. [58], and Constantinides et al. [73]. This might be attributed 

to the high amount of circuitry necessary for an entire 

nonrecursive implementation, or to the fact that wave digital 

filters [61] could not have or have not been familiar to those 

C- 

authors. The only approaches published up to now requiring 

negligible additional hardware for assuring absolute stability 

Fig. 29. Closed loop due to nonideal two--+re/four-wire transitions under looped conditions were proposed by Fettweis [59], [60] 

[l],[651. 

and most recently by Kurth et al. [8 l], [82]. 

The next column of Table I1 is related to the absolute value 

or, to which extent, additional circuitry, such as a separate 

translator, is required. 

of the group delay measured at the analog ports of a transmultiplexer, 

the digital ports being looped (cf., Table I). As 

stated above, the basic group delay may not exceed 3 ms. For 

this reason, all those transmultiplexer implementations exclusively 

using linear phase FIR filters are generally not applicable. 

However, within the great majority of these approaches the 

linear phase filters could be replaced by minimum phase filters 

at the expense of a generally lower computational efficiency. 

This does not hold for the proposal of Constantinides et al. 

[41], since fast convolution techniques being based on block 

processing are applied for filtering (cf., [23]). Moreover, it 

must be noted that substantial delay may arise in transmultiplexer 

realizations with an inherent multiple-way modulation 

scheme utilizing DFT, DCT, or related signal processors. 

From the next column of Table I1 the operation rate per 

channel, or the multiplication (M) and addition (A) rates per 

channel can be seen, respectively. Here, the algorithms of 

Bellanger et al. [42]-[45], and Narasimha et al. [50], [86] 

being based on a multiple-way modulation scheme are, by


far, most efficient. According to the subsequent column of 

Table I1 referring to the degree of modularity, the control 

circuitry required for these approaches is, on the contrary, 

expected to be rather complex. This is confirmed by Bellanger 

et al. [70] giving a figure for their amount of transmultiplexer 

overhead circuitry of up to 30 percent of the total hardware. 

Moreover, the transmultiplexer methods with an inherent 

multiple-way modulation scheme generally require the longest 

words for coefficient representation [70]. Hence, these approaches 

and, for instance, the proposal by Fettweis [59]- 

necessitating substantially shorter coefficients' wordlengths at 

a higher multiplication rate-can only be compared with each 

other on the bit level. However, such and other more detailed 

discussion of hardware aspects is beyond the scope of this 

paper. The interested reader is referred to [70], [71 l. 

Conce~ng the degree of modularity, those transmultiplexer 

approaches needing only few interconnections between different 

modules and which exhibit a regular overall structure are 

classified to be highly modular. Hence, all methods applying 

a highly meshed signal processor and a filter bank with all 

filter coefficients being generally different are of a low degree 

of modularity. The poorest specimens of this sort seem to be 

the sophisticated transmultiplexer approaches of Peled et al. 

[55] and Molo [74]. On the contrary, a bank of Weaver 

modulators [ 1 S], [S l] -[53], and some multistage modulation 

methods [36], [56], [59], [73] exhibit a moderate or even 

higher degree of modularity. It is felt, that the approach of 

Kurth et al. [81], [82] shows the highest degree of modularity 

compared with all other methods of Table 11. Here, even the 

implementation of channel filters is highly modular. 

Finally, the last two columns of Table I1 are essentially selfevident. 

Suffice to say that a 12 (24)-channel transmultiplexer 

directly generating the FDM signals in the desired group band 

frequency range actually requires an additional (analog) group 

band to supergroup (SG) convertor, if it is to be used at the 

supergroup level. Furthermore, separate processing of outof-band 

signaling pulses is necessary for all North American 

transmultiplexer systems, since these are designed for in-band 

signaling. Although being based on the same 24-channel PCM 

frame as in North America, Japanese domestic FDM systems 

apply out-of-band signaling (cf., [8, table 111). 

VIII. CONCLUSION 

With this report an attempt was made to present a comprehensive 

survey of the currently known digital transmultiplexer 

methods. In order to cope with this task, the necessary theoretical 

background, such as analog and digital modulation 

techniques, and the complex of sample rate alteration (i.e., 

decimation and interpolation), is contained. Within the main 

section of this paper, the various transmultiplexing methods 

were classified into four categories: bandpass fdter-bank 

method, low-pass filter-bank method, Weaver structure method, 

and multistage modulation methods. Furthermore, on the 

basis of a list of criteria given in the preceding section, the 

individual transmultiplexer approaches were compared with 

each other. 

When concluding, we first pick up again the two most important 

items of Table 11: Absolute stability under looped 

conditions, and the CCITT Recommendation concerning the 

absolute value of the group delay measured at the analog ports 

of a transmultiplexer, the digital ports being looped, which 

may not exceed 3 ms. Refening to stability in a looped arrangement 

(Fig. 29), we follow the derivations of Fettweis 

et al. [l], [651 rigorously. (It must be noted, however, that 

the resulting conditions for guaranteeing absolute stability 

under any adverse circumstances are sufficient but not necessary.) 

As a consequence of this assumption, we are forced to 

drop almost all, even computationally most efficient transmultiplexer 

approaches, such as 1441, [SO], due to potential 

stability problems. At present, the only exceptions are the 

multistage modulation method proposed by Fettweis [59], 

1601 and the per-channel approach by Kurth et al. [811,1821 . 

Certainly, the algorithms published by Kurth [S21 and Kao 

[53], Freeny et al. [IS], [Sl], Singh et al. [54], Del Re 1831, 

Tsuda et al. [56], [58], and Constantinides et al. [73] can 

also be used, provided that the implementation of these approaches 

is restricted to the sole application of wave digital 

andtor nomecursive filters. From these, however, only the 

two latter ones deserve further interest, since the other approaches 

mentioned here are relatively inefficient (cf., Table 

11). Morebver, if the Weaver modulator bank were extended to 

60 channels, this gap would still get wider due to the increased 

sampling rate alteration ratio. 

The following comparison can, thus, be limited to the singleway 

modulation schemes of Fettweis [59], [60] and Kurth 

et al. 1811, 1821, and the two-way modulation schemes of 

Tsuda et al. [%l, 1.581 and Constantinides et al. [73]. The 

latter, however, applies recursive interpolators throughout the 

whole transmultiplexer. As a consequence, the sole use of 

wave digital andtor nonrecursive fdters would lead to a fundamental 

alteration of this approach: More or less, it would 

result in a new approach. Hence, we preclude this method, 

which is nevertheless very interesting due to its low computational 

load, from the following final discussion. 

Next, if the first most complex filter at the PCM end of the 

Tsuda approach were realized as a wave digital filter, the multiplication 

rate per channel according to Table I1 would essenti- 

ally be retained. However, in order to guarantee absolute 

stability under looped conditions, the additional amount of 

circuitry required for this two-way modulation scheme is expected 

to be substantially higher than that necessary for 

Fettweis' single-way scheme [ 11. 

Furthermore, in Tsuda's 

approach the minimum absolute value of the group delay of 

the multistage filter cascade of about 2.6 ms is not very far 

removed from the maximum allowable figure of 3 ms. A wider 

margin can, however, only be gained with the application of 

minimum phase FIR instead of linear phase halfband filters at 

the expense of a lower computational efficiency (cf., [57]). 

Finally, it must be noted that an additional analog frequency 

convertor has to be provided. On the other hand, the singleway 

modulation scheme of Fettweis is less regular (less modular) 

than Tsuda's transmultiplexer approach. Therefore, it 

depends essentially on the reader's particular weighting of the 

(optional) criteria discussed in Section VII, which of these two 

transmultiplexer approaches is preferable. 

Most recently, the discussion on different transmultiplexing 

methods has been promoted by the resumption of one of the 

first ideas in transmultiplexing history [l51 by Kurth et al. 

[8 l], 1821 : The direct extraction of the desired frequency 

band from the overall wide-band spectrum by a single interpolation 

or decimation filter per channel, respectively. Due to 

logarithmic processing this proposal is memory oriented. Since 

no multiplication at all must be carried out, the amount of 

memory seems to be extremely high: Between any multiply 

and add operations of the direct form minimum-phase FIR 

filter used, the logarithmic number representation has to be

SCHEUERMANN AND GOCKLER: DIGITAL TRANSMULTIPLEXING METHODS 1449 

converted to linear format. From this it follows immediately 

that all convertors with the exception of the p-law/logarithmic 

look-up table at each filter input are identical. Consequently, 

these convertor memories can efficiently be time-multiplexed 

within a channel filter and/or across the channels, since very 

fast ROM's are readily available. In conclusion, this approach 

taking full advantage of today's fast advancing memory technology 

seems to be one of the most promising transmultiplexing 

methods for the future. This approach is particularly 

striking due to its inherent high degree of modularity, requiring 

only one type of a direct form FIR interpolation filter with 

time-varying coefficients (cf., [921) which, by itself, is highly 

regular. With this approach the widespread expectation "the 

simpler, the better" seems to be best met. 

Nevertheless, in order to stimulate the further discussion on 

efficient, absolutely stable, and modular transmultiplexing 

methods, the reader is referred to a .forthcoming paper on a 

novel 60-channel transmultiplexer approach being based on 

a single-way multistage conversion scheme using directional 

filters [72], [go]. Furthermore, an extension of the proposal 

by Kurth et al. [8 1 ] with reduced computational load is presently 

being investigated [91]. 

The authors are greatly indebted to Prof. Fettweis and Dr. L. 

Gazsi, Ruhr Universitat Bochum, to Prof. Schiissler, Dr. U. 

Heute and Dr. P. Vary, Universitat Erlangen-Niirnberg, and to 

Dr. R. Lagadec and Dr. D. Pelloni, formerly ETH Zurich, for 

various stimulating discussions on the topic treated in this 

paper. Furthermore, we would like to thank Dr. E. Gleissner 

for carefully reading the initial manuscript and providing valuable 

constructive criticisms. 

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PROCEEDINGS OF THE IEEE, VOL. 70, NO. 7, JULY 1982 

Then, the real output signals of the DFT processor can most easily be 

derived, if the samples of the complex signals (26) are taken at odd 

multiples of vT/2 (instead of integer multiples of vT) leading to a 

W-point odd-time inverse discrete Fourier transfo~m (IODFT)~ 1431 

Using (26), (27) may b'e factored into 

Correction to "A Comprehensive Survey of Digital 

Transmultiplexing Methods" 

HELMUT SCHEUERMANN AND HEINZ GOCKLER 

In the above-titled survey paper1 the basic derivation of the lowpass 

frlter-bank method at the beginning of section VI-B is somewhat 

erroneous. The correct derivation is given leading to a minor modification 

of Fig. 18.' In addition, two errors are corrected as follows: 

B. Low-Pass Filter-Bank Method 

If one succeeds in obtaining real signals at the output of the Fourier 

processor, processing of "complex" signals in the polyphase network 

can be avoided. This is achieved if the complex input signals to a W- 

point DFT processor exhibit Herrnitian symmetry 

~ppiying the variable transformation K = 2N - r - 1, the second tern 

of (28) yields 

Thus there results from (28), if in its fust term r is replaced by K 

Manuscript received January 20, 1982. 

The authors are with AEG-Telefunken Nachrichtentechnik GmbH, 

Advanced Development Department, D-7150 Backnang, Germany. 

'H, Scheuermann and H. GocMer, Proc. ZEEE, vol. 69, pp. 1419- 

1450, NOV. 1981. 

a G. Bonnerot and M. Bellanger, "Odd-time odd-frequency discrete 

Fourier transform for symmetric real-valued series," Proc. ZEEE, vol. 

64, pp. 392-393, Mar. 1976.

PROCEEDINGS OF THE IEEE, VOL.,70, NO. 7, JULY 1982 

From this, the z-transform of the transmultiplexer output sequence is 

obtained in the natural sequential order of summation 

and, accordingly, by interchanging the summation order 

As a consequence, the IDFT processor in Fig. lgl must be replaced by 

a 2N-point IODFT processor. 

Finally, two errors are indicated: The fust sentence of the last paragraph 

of p. 1432 should read: "Still another transmultiplexer, proposed 

by Tomlinson and Wong [48], [49], works according to the low-pass 

filter-bank method." Second, the author of 1631 is A. Fettweis.

HELMUT SCHEUERMANN AND HElNZ GOCKLER - der ...

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