B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

122 ANALYSIS AND TRANSMISSION OF SIGNALS
sampling interval Jo = 1 /To of G(f). Moreover, if G(f) is not band-limited, we need to truncate
it. This will cause an additional error in the computation of gk . By increasing the truncation
bandwidth, we can reduce this error. In practice, (tapered) window functions are often used
for truncation 5 in order to reduce the severity of some problems caused by straight truncation
(also known as rectangular windowing).
Because G q
is No-periodic, we need to determine the values of G q
over any one period.
It is customary to determine G q
over the range (0, No - 1) rather than over the range
(-No/2, No/2 - 1). The identical remark applies to gk .
Choice of Ts, To, and No
In DFf computation, we first need to select suitable values for No, Ts, and To . For this purpose
we should first decide on B, the essential bandwidth of g(t). From Fig. 3.39d, it is clear that the
spectral overlapping (aliasing) occurs at the frequency fs/2 Hz. This spectral overlapping may
also be viewed as the spectrum beyondfs/2 folding back atfs/2. Hence, this frequency is also
called the folding frequency. If the folding frequency is chosen such that the spectrum G(f ) is
negligible beyond the folding frequency, aliasing (the spectral overlapping) is not significant.
Hence, the folding frequency should at least be equal to the highest significant frequency, that
is, the freq1,1ency beyond which G(f) is negligible. We shall call this frequency the essential
bandwidth B (in hertz). If g(t) is band-limited, then clearly, its bandwidth is identical to the
essential bandwidth. Thus,
t > B Hz
2 -
(3. 105a)
Moreover, the sampling interval Ts = 1/f, [Eq. (3. 104)]. Hence,
T. <
a - 2B -
(3. 105b)
Once we pick B, we can choose T, according to Eq. (3. 105b). Also,
1
fo =- (3. 106)
To
where Jo is the frequency resolution [separation between samples of G(f)]. Hence, iffo is
given, we can pick To according to Eq. (3. 106). Knowing To and T s , we determine No from
To
No = ­ Ts
(3. 107)
In general, if the signal is time-limited, G(f) is not band-limited, and there is aliasing in
the computation of G q. To reduce the aliasing effect, we need to increase the folding frequency;
that is, we must reduce T, (the sampling interval) as much as is practicable. If the signal is
band-limited, g(t) is not time-limited, and there is aliasing (overlapping) in the computation
of gk. To reduce this aliasing, we need to increase To, the period of gk. This results in reducing
the frequency sampling interval fo (in hertz). In either case (reducing Ts in the time-limited
case or increasing To in the band-limited case), for higher accuracy, we need to increase the
number of samples No because No = To/Ts . There are also signals that are neither time-limited
nor band-limited. In such cases, we need to reduce T i and increase To.
Points of Discontinuity
If g (t) has a jump discontinuity at a sampling point, the sample value should be taken as the
average of the values on the two sides of the discontinuity because the Fourier representation
at a point of discontinuity converges to the average value.