Stability is perhaps the most important property of real systems. When a system is unstable a number of its operating parameters may change freely or go without bound, or (for computer pregrammes) give inconsistent results.
For DSP (or DTSP) systems the definition of stability is as follows: The system is stable when with respect to a bounded input it gives a bounded output. This stability criterion is called bounded-input bounded-output (BIBO). Mathematically:
|
x(n)
|≤
M
x
<∞ ⇒ |
y(n)≤
M
y
<∞
|
|
x(n)
|≤
M
x
<∞ ⇒ |
y(n)≤
M
y
<∞
|
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaemaabaGaamiEaiaacIcacaWGUbGaaiykaaGaay5bSlaawIa7aiabgsMiJkaad2eadaWgaaWcbaGaamiEaaqabaGccqGH8aapcqGHEisPcaaMf8UaeyO0H4TaaGzbVpaaemaabaGaamyEaiaacIcacaWGUbGaaiykaiabgsMiJkaad2eadaWgaaWcbaGaamyEaaqabaGccqGH8aapcqGHEisPaiaawEa7caGLiWoaaaa@548C@
Now we derive the condition of stability imposed on impulse response. Starting from the convolution summation
y(n)=x(n)*h(n)=
∑
k=−∞
+∞
x(n)h(n−k)
y(n)=x(n)*h(n)=
∑
k=−∞
+∞
x(n)h(n−k)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bGaaiikaiaad6gacaGGPaGaaiOkaiaadIgacaGGOaGaamOBaiaacMcacqGH9aqpdaaeWbqaaiaadIhacaGGOaGaamOBaiaacMcacaWGObGaaiikaiaad6gacqGHsislcaWGRbGaaiykaaWcbaGaam4Aaiabg2da9iabgkHiTiabg6HiLcqaaiabgUcaRiabg6HiLcqdcqGHris5aaaa@53AB@
Let's take the absolute value of both sides:
|
y(n)
|=|
∑
k=−∞
∞
x(n)h(n−k)
|≤
∑
k=−∞
∞
|
x(n)
|
|
h(n−k)
|=|
x(n)
|
∑
k=−∞
∞
|
h(n−k)
|
|
y(n)
|=|
∑
k=−∞
∞
x(n)h(n−k)
|≤
∑
k=−∞
∞
|
x(n)
|
|
h(n−k)
|=|
x(n)
|
∑
k=−∞
∞
|
h(n−k)
|
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@80FA@
then for finite
∣x(n)∣,∣y(n)∣∣x(n)∣,∣y(n)∣ size 12{ lline x \( n \) rline , lline y \( n \) rline } {}is finite if
∑
k=−∞
∞
|
h(k)
|
<∞
∑
k=−∞
∞
|
h(k)
|
<∞
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaWaaqWaaeaacaWGObGaaiikaiaadUgacaGGPaaacaGLhWUaayjcSdaaleaacaWGRbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdGccqGH8aapcqGHEisPaaa@46BB@
Since k is a dummy variable we can change it to n and write the condition as
∑
n=−∞
∞
|
h(n)
|
<∞ (Condition of stability BIBO)
∑
n=−∞
∞
|
h(n)
|
<∞ (Condition of stability BIBO)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaWaaqWaaeaacaWGObGaaiikaiaad6gacaGGPaaacaGLhWUaayjcSdaaleaacaWGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdGccqGH8aapcqGHEisPcaaMf8UaaGzbVlaaywW7caGGOaGaam4qaiaad+gacaWGUbGaamizaiaadMgacaWG0bGaamyAaiaad+gacaWGUbGaaGzbVlaad+gacaWGMbGaaGzbVlaadohacaWG0bGaamyyaiaadkgacaWGPbGaamiBaiaadMgacaWG0bGaamyEaiaaywW7caWGcbGaamysaiaadkeacaWGpbGaaiykaaaa@6753@
(1)
That is, the impulse response is obsolutely summable . FIR systems are mostly stable , whereas as for IIR systems the stablity requires the impulse response decays fast enough with time.
Example 1 A LTI system has impulse response
h(n)=
a
n
n≥0
=
b
n
n<0
h(n)=
a
n
n≥0
=
b
n
n<0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaGaamiAaiaacIcacaWGUbGaaiykaiabg2da9iaadggadaahaaWcbeqaaiaad6gaaaGccaaMf8UaaGzbVlaad6gacqGHLjYScaaIWaaabaaabaGaaGzbVlaaywW7cqGH9aqpcaWGIbWaaWbaaSqabeaacaWGUbaaaOGaaGzbVlaaywW7caWGUbGaeyipaWJaaGimaaaaaa@4EC3@
Find the condition for stability.
Solution
The overall impulse response consists of a causal part and a noncausal one. The condition of stability is
∑
n=−∞
+∞
|
h(n)
|
=
∑
n=0
+∞
| a |
n
+
∑
n=−∞
−1
| b |
n
<∞
∑
n=−∞
+∞
|
h(n)
|
=
∑
n=0
+∞
| a |
n
+
∑
n=−∞
−1
| b |
n
<∞
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaaqahabaWaaqWaaeaacaWGObGaaiikaiaad6gacaGGPaaacaGLhWUaayjcSdaaleaacaWGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaey4kaSIaeyOhIukaniabggHiLdGccqGH9aqpdaaeWbqaamaaemaabaGaamyyaaGaay5bSlaawIa7amaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaGimaaqaaiabgUcaRiabg6HiLcqdcqGHris5aOGaey4kaSYaaabCaeaadaabdaqaaiaadkgaaiaawEa7caGLiWoadaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2da9iabgkHiTiabg6HiLcqaaiabgkHiTiaaigdaa0GaeyyeIuoakiabgYda8iabg6HiLcaa@6363@
First
∑
n
=
0
∞
∣
a
n
∣
=
1
+
∣
a
∣
+
∣
a
∣
2
+
.
.
.
<
∞
∑
n
=
0
∞
∣
a
n
∣
=
1
+
∣
a
∣
+
∣
a
∣
2
+
.
.
.
<
∞
size 12{ Sum cSub { size 8{n=0} } cSup { size 8{ infinity } } { lline a rSup { size 8{n} } rline =1+ lline a rline + lline a rline rSup { size 8{2} } + "." "." "." < infinity } } {}
Applying the formula of infinite geometric series (
Equation) will lead to the condition
∣a∣<1∣a∣<1 size 12{ lline a rline <1} {}.
Now
∑
n=−∞
−1
| b |
n
=
∑
n=1
∞
1
| b |
n
=
1
| b |
[
1+
1
| b |
+
1
| b |
2
+...
]
=
1
| b |
1
1−
1
| b |
,
1
| b |
<1
∑
n=−∞
−1
| b |
n
=
∑
n=1
∞
1
| b |
n
=
1
| b |
[
1+
1
| b |
+
1
| b |
2
+...
]
=
1
| b |
1
1−
1
| b |
,
1
| b |
<1
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8AED@
The condition is
1∣b∣1∣b∣ size 12{ { {1} over { lline b rline } } } {} < 1 or
∣b∣>1∣b∣>1 size 12{ lline b rline >1} {}. The overall condition is
∣a∣<1∣a∣<1 size 12{ lline a rline <1} {} and
∣b∣>1∣b∣>1 size 12{ lline b rline >1} {}
