7.1 Chapter Summary . . . . 111 7.2 Shifting . . . . 111 7.3 Shift-Invariant Discrete Linear Systems . . . . 112 7.4 The Delta Sequence . . . . 112 7.5 The Discrete Impulse Response . . . . 112 7.6 The Discrete Transfer Function . . . . 113 7.7 Using Fourier Series . . . . 114 7.8 The Multiplication Theorem for Convolution . . . . 114 7.9 The Three-Point Moving Average . . . . 115 7.10 Autocorrelation . . . . 116 7.11 Stable Systems . . . . 117 7.12 Causal Filters . . . . 118
7.1 Chapter Summary
Many textbooks on signal processing present filters in the context of infinite sequences. Although infinite sequences are no more realistic than functionsf(t) defined for all timest, they do simplify somewhat the discus-sion of filtering, particularly when it comes to the impulse response and to random signals. Systems that have as input and output infinite sequences are calleddiscrete systems.
7.2 Shifting
We denote byf ={fn}∞n=−∞ an infinite sequence. For a fixed integer k, the system that accepts f as input and produces as output the shifted sequenceh={hn=fn−k}is denotedSk; therefore, we write h=Skf.
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7.3 Shift-Invariant Discrete Linear Systems
A discrete systemT islinearif
T(af1+bf2) =aT(f1) +bT(f2),
for any infinite sequencesf1 andf2 and scalars aandb. As previously, a systemT is shift-invariant ifT Sk =SkT. This means that if input f has outputh, then inputSkf has outputSkh; shifting the input bykjust shifts the output byk.
7.4 The Delta Sequence
Thedelta sequenceδ={δn} hasδ0= 1 andδn= 0, fornnot equal to zero. Then Sk(δ) is the sequence Sk(δ) ={δn−k}. For any sequence f we have
fn= ∞ m=−∞
fmδn−m= ∞ m=−∞
δmfn−m.
This means that we can write the sequence f as an infinite sum of the sequencesSmδ:
f =
∞ m=−∞
fmSm(δ). (7.1)
As in the continuous case, we use the delta sequence to understand better how a shift-invariant discrete linear systemT works.
7.5 The Discrete Impulse Response
We let δ be the input to the shift-invariant discrete linear system T, and denote the output sequence byg=T(δ). Now, for any input sequence
f withh=T(f), we writef using Equation (7.1), so that
for eachn. Equation (7.2) is the definition of discrete convolution or the convolution of sequences. This tells us that the output sequenceh=T(f) is the convolution of the input sequencef with the impulse-response sequence g; that is,h=T(f) =f∗g.
7.6 The Discrete Transfer Function
Associated with each ω in the interval [0,2π) we have the sequence eω ={e−inω}∞n=−∞; the minus sign in the exponent is just for notational convenience later. What happens when we let f =eω be the input to the systemT? The output sequencehwill be the convolution of the sequence eωwith the sequenceg; that is,
This tells us that when eω is the input, the output is a multiple of the input; the “frequency” ωhas not changed, but the multiplication byG(ω) can alter the amplitude and phase of the complex-exponential sequence.
Notice that Equation (7.3) is the definition of the Fourier series asso-ciated with the sequenceg viewed as a sequence of Fourier coefficients. It follows that, once we have the functionG(ω), we can recapture the original gn from the formula for Fourier coefficients:
gn= 1
It follows that we can write
f = 1
2π 2π
0
F(ω)eωdω. (7.4)
We interpret this as saying that the sequence f is a superposition of the individual sequenceseω, with coefficientsF(ω).
7.8 The Multiplication Theorem for Convolution
Now considerf as the input to the systemT, withh=T(f) as output.
Using Equation (7.4), we can write h=T(f) =T
But, applying Equation (7.4) toh, we have
h= 1
2π 2π
0
H(ω)eωdω.
It follows that H(ω) = F(ω)G(ω), which is analogous to what we found in the case of continuous systems. This tells us that the system T works by multiplying the functionF(ω) associated with the input by the transfer function G(ω), to get the function H(ω) associated with the output h= T(f). In the next section we give an example.
7.9 The Three-Point Moving Average
We consider now the linear, shift-invariant systemT that performs the three-point moving averageoperation on any input sequence. Letf be any input sequence. Then the output sequence ishwith
hn= 1
3(fn−1+fn+fn+1).
The impulse-response sequence isg withg−1=g0=g1 = 13, andgn = 0, otherwise.
To illustrate, for the input sequence with fn = 1 for alln, the output ishn = 1 for alln. For the input sequence
f ={...,3,0,0,3,0,0, ...},
the output h is again the sequence hn = 1 for all n. If our input is the difference of the previous two input sequences, that is, the input is {...,2,−1,−1,2,−1,−1, ...}, then the output is the sequence with all en-tries equal to zero.
The transfer functionG(ω) is G(ω) = 1
3(eiω+ 1 +e−iω) = 1
3(1 + 2 cosω).
The functionG(ω) has a zero when cosω =−12, or whenω= 23π orω= 43π. Notice that the sequence given by
fn=
ei2π3n+e−i2π3n
= 2 cos2π 3 n
is the sequence {...,2,−1,−1,2,−1,−1, ...}, which, as we have just seen, has as its output the zero sequence. We can say that the reason the output
is zero is that the transfer function has a zero atω= 23π and at ω=43π =
−2π
3 . Those complex-exponential components of the input sequence that correspond to values ofω where G(ω) = 0 will be removed in the output.
This is a useful role that filtering can play; we cannull out an undesired complex-exponential component of an input signal by designing G(ω) to have a root at its frequencyω.
7.10 Autocorrelation
If we take the input to our convolution filter to be the sequencefrelated to the impulse-response sequence by
fn=g−n, then the output sequence ishwith entries
hn= +∞
k=−∞
gkgk−n
andH(ω) =|G(ω)|2. The sequencehis called theautocorrelation sequence forgand|G(ω)|2 is thepower spectrum ofg.
Autocorrelation sequences have special properties not shared with or-dinary sequences, as the exercise below shows. The Cauchy Inequality is valid for infinite sequences: with the length ofgdefined by
g= +∞
n=−∞
|gn|21/2
and the inner product of any sequencesf andg given by f, g =
+∞
n=−∞
fngn, we have
|f, g | ≤ f g,
with equality if and only ifg is a constant multiple off.
Ex. 7.1 Let hbe the autocorrelation sequence for g. Show thath−n =hn andh0≥ |hn| for alln.
7.11 Stable Systems
An infinite sequence f ={fn} is called boundedif there is a constant A >0 such that|fn| ≤A, for alln. The shift-invariant linear system with impulse-response sequenceg=T(δ) is said to bestable[120] if the output sequenceh={hn}is bounded whenever the input sequencef ={fn}is. In Exercise 7.2 below we ask the reader to prove that, in order for the system to be stable, it is both necessary and sufficient that
∞ n=−∞
|gn|<+∞.
Given a doubly infinite sequence,g ={gn}+∞n=−∞, we associate with g its z-transform, the function of the complex variablez given by
G(z) =
+∞
n=−∞
gnz−n.
Doubly infinite series of this form are called Laurent series and occur in the representation of functions analytic in an annulus. Note that if we take z =e−iω then G(z) becomes G(ω) as defined by Equation (7.3). The z-transform is a somewhat more flexible tool in that we are not restricted to those sequencesgfor which the z-transform is defined forz=e−iω. Ex. 7.2 Show that the shift-invariant linear system with impulse-response sequence g is stable if and only if
+∞
n=−∞
|gn|<+∞. Hint: If, on the contrary,
+∞
n=−∞
|gn|= +∞, consider as input the bounded sequencef with fn=g−n/|g−n| and show that the outputh0= +∞.
Ex. 7.3 Consider the linear system determined by the sequence g0 = 2, gn = (12)|n|, for n = 0. Show that this system is stable. Calculate the z-transform of{gn} and determine its region of convergence.
7.12 Causal Filters
The shift-invariant linear system with impulse-response sequence g is said to be a causal system if the sequence {gn} is itself causal; that is, gn = 0 for n < 0. For causal systems the value of the output atn, that is,hn, depends only on those input valuesfm form≤n. When the input is a time series, this says that the value of the output at any given time depends only on the value of the inputs up to that time, and not on future values of the input sequence. A number of important filters, such as band-limiting filters, are not causal and have to be approximated by causal filters to operate in real time.
Ex. 7.4 Show that the function G(z) = (z−z0)−1 is the z-transform of a causal sequence g, wherez0 is a fixed complex number. What is the region of convergence? Show that the resulting linear system is stable if and only if|z0|<1.