U. Glaeser

The output of a linear system having an impulse response h[k] to an input x[k] is denoted y[k]=h[k] ×
x[k], where y[k] is defined by the discrete-time linear convolution sum
Computing the convolution sum is rare. Instead, a linear system is generally analyzed using simulation,
emulation, or the z-transform. The convolution theorem states that the linear convolution y[k] = h[k] ×
∞
x[k] of a z-transformable impulse response h[k] (i.e., H(z) = Σk=−∞ h[ k]z)
and input x[k] (i.e., X(z)
= ), is given by the inverse z-transform of the product Y(z)=H(z)X(z). This method is only
viable in instances where the z-transform of h[k] and x[k] have been precomputed or tabled, and the
inverse z-transform of Y[z] can be computed. While H(z) is generally known, most real signals are
arbitrary and possibly noise contaminated, making the general availability of X(z) questionable. Nevertheless,
the importance of this equation has resulted in the elements being given specific titles and
meaning. The z-transform of the impulse response h[k], namely H(z), is called the system’s transfer
function and has the general form
k –
∞
Σk=−∞ x[ k]z
k –
The filter’s poles (p m) and zeros (z m) are the roots of D(z) = 0 and N(z) = 0, respectively. The system’s
steady-state frequency response can be determined by evaluating the transfer function H(z) along the
trajectory z = e jϖ , where ϖ ∈[−π, π] which represents a normalized baseband frequency range [−f s /2, f s /2]
(±Nyquist frequency). Specifically, the frequency response of a system in magnitude-phase form is H(e jϖ ) =
|H(e jϖ )| ∠ φ(e jϖ ).
24.3 Digital Filters
Transfer functions, when implemented in the time-domain, result in digital filters. The attributes of a digital
filter can be specified in the time- or frequency-domain, or both. Digital filters can be grouped into three
broad classes called finite impulse response (FIR), infinite impulse response (IIR), and multirate filters.
Finite Impulse Response (FIR) Filters
An FIR filter possesses an impulse response that persists only for a finite number of sample values. The
impulse response of an Nth order FIR is given by h[k] = {h[0],…, h N−1[k]}, and in the z-transform domain
by H(z) = Σh iz −i , i ∈[0, N − 1]. One of the attributes of an FIR is its simplicity, consisting of a string of
multiply-accumulations (MACs), and shifts registers. The steady-state frequency response of an FIR H(z)
is given by H(e jϖ ) = |H(e jϖ )| ∠φ(e jϖ ). A system is said to possess a linear phase response if φ(e jϖ ) = αϖ + β
(i.e., linear in frequency). Linear phase filters are important in a number of applications including (1)
synchronizing phase modulated data streams, (2) anti-aliasing filters placed in front of signal phase
sensitive analysis subsystems (e.g., FFT), and (3) use in phase sensitive applications (e.g., image processing).
Linear phase filtering can be guaranteed whenever the coefficients of an N th order FIR are symmetrically
distributed about the filter’s mid-point L = (N−1)/2 (i.e., h i = ±h N−i, i = 0,…,L). The resulting
phase response satisfies the linear phase equation ∠φ(ϖ) = −Lϖ + {0,±π}. Another important phase
response measure is called the group delay, given by τ g = −dφ(e jϖ )/dϖ. For a linear phase FIR, τ g = L,
which indicates that the filter propagation delay is always L clock cycles regardless of the input signal
frequency.
© 2001 by CRC Press LLC
N
y[ k]
= – ∑ amy[ k– m]
+ ∑ bmx[ k– m].
m=1
M
m=0
H( z)
Y( z)
⁄ X( z)
bmz m –
–
= = =
M
∑ a / mz m
∑
m=0
N
m=0
N( z)/D(
z).