IMPULSE RESPONSE1.22007/11/04 19:57:38 US/Central2008/07/02 02:22:38.959 GMT-5PhuongHuuNguyennhphuong@hcmuns.edu.vnPhuongHuuNguyennhphuong@hcmuns.edu.vnWe are now looking for a more consise way to characterize discrete-time systems.
Unit sample
Let’s recall the unit sample (also called unit impulse) signal ()δ(n)=1,n=0=0,n≠0alignl { stack {
size 12{δ \( n \) =1, matrix {
{} # {} # {}
} n=0} {} #
size 12{ matrix {
{} # {}
} =0, matrix {
{} # {} # {}
} n <> 0} {}
} } {}When the sample is shifted to time index (or sample) k in the future
(k>0) size 12{ \( k>0 \) } {} the signal is
δ(n−k)=1,n=k0,n≠kalignl { stack {
size 12{δ \( n - k \) =1, matrix {
{} # {}
} n=k} {} #
size 12{ matrix {
{} # {} # {} # {}
} 0, matrix {
{} # {}
} n <> k} {}
} } {}When the sample is shifted to the past at index –k
(k>0) size 12{ \( k>0 \) } {}, the signal is
δ(n+k) size 12{δ \( n+k \) } {}δ(n+k)=1,n=−k0,n≠−kalignl { stack {
size 12{δ \( n+k \) =1, matrix {
{} # {}
} n= - k} {} #
size 12{ matrix {
{} # {} # {} # 0, matrix {
{} # {}
} {}
} n <> - k} {}
} } {}Now,let’s express a signal in terms of unit samples. In the value of
x(n) size 12{x \( n \) } {} at
n=1 size 12{n=1} {} is 3, so we can write ( Equation )x(n=1)=x(1)δ(n−1)=3x(1)=3 size 12{x \( n=1 \) =x \( 1 \) matrix {
{} # {}
} δ \( n - 1 \) =3 matrix {
{} # {}
} x \( 1 \) =3} {}
An example signal
Similarly at
n=2 size 12{n=2} {}x(n=2)=x(n)δ(n−2)=2x(1)=2 size 12{x \( n=2 \) =x \( n \) matrix {
{} # {}
} δ \( n - 2 \) =2 matrix {
{} # {}
} x \( 1 \) =2} {}Thus a signal
x(n) size 12{x \( n \) } {} can be expressed asx(n)=∑k=−∞∞x(k)δ(n−k) size 12{x \( n \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( k \) δ \( n - k \) } } {}Impulse responseImpulse response (or impulsive response) of a discrete-time (or digital) system is defined as the output (response), denoted by
h(n) size 12{h \( n \) } {}, when the input is an unit sample
δ(n) size 12{δ \( n \) } {}. The impulse response may be real or complex, but usually assumed real. is an example.
Definition and example of impulse response
FIR and IIR systemsWhen excited by an unit sample
δ(n) size 12{δ \( n \) } {}, the impulse response
h(n) size 12{h \( n \) } {}of a system may last a finite duration, or forever even before the input is applied. In the former case the system is Finite Duration Impulse Response (FIR), and in the latter case the system is Infinite Duration Impulse Response (IIR). Many authors do not include the word duration in the names. It’s just a matter of choice.Alternatively, the systems can be classfied as Recursive or Nonrecursive instead of FIR or IIR. We’ll see the difference later. Causality (see section )of a system is reffected on its impulse resopnse: For causal systems
h(n)=0 size 12{h \( n \) =0} {} at
n<0 size 12{n<0} {} (or
n≤−1 size 12{n <= - 1} {}), otherwise they are noncausal. Both systems of Figure are noncausal.
Example of systems (a) FIR, (b) IIR
Derive the impulse response from the difference equationFrom the definition of impulse response we can apply an unit sample to the system concerned and obtain the output experimentally, which is the impulse response. Otherwise we derive it from the difference equation as presented here. There are still other ways to obtain the impulse response.Find the impulse response of system whose input-output signal difference equation is given by
y(n)=0.8y(n−1)+x(n) size 12{y \( n \) =0 "." 8y \( n - 1 \) +x \( n \) } {} SolutionReplacing
x(n) size 12{x \( n \) } {} by
δ(n) size 12{δ \( n \) } {} then
y(n) size 12{y \( n \) } {} is just
h(n) size 12{h \( n \) } {}:h(n)=0.8h(n−1)+δ(n) size 12{h \( n \) =0 "." 8h \( n - 1 \) +δ \( n \) } {}Remember that
δ(n)=1 size 12{δ \( n \) =1} {} at
n=0 size 12{n=0} {}, otherwise zero, and assume a causal system, i.e.
h(n)=0 size 12{h \( n \) =0} {} for
n<0 size 12{n<0} {}, we haveh(0)=0.8h(−1)+δ(0)=1h(1)=0.8h(0)+δ(1)=0.8h(2)=0.8h(1)+δ(2)=0.82h(3)=0.8h(2)+δ(3)=0.83....h(n)=0.8nu(n)alignl { stack {
size 12{h \( 0 \) =0 "." 8h \( - 1 \) +δ \( 0 \) =1} {} #
size 12{h \( 1 \) =0 "." 8h \( 0 \) +δ \( 1 \) =0 "." 8} {} #
size 12{h \( 2 \) =0 "." 8h \( 1 \) +δ \( 2 \) =0 "." 8 rSup { size 8{2} } } {} #
h \( 3 \) =0 "." 8h \( 2 \) +δ \( 3 \) =0 "." 8 rSup { size 8{3} } {} #
"." "." "." "." {} #
h \( n \) =0 "." 8 rSup { size 8{n} } u \( n \) {}
} } {}The system is IIR and stable (since
h(n) size 12{h \( n \) } {} converges). Usually we don’t get the result in closed form as above. See also Example Derive the difference equation from the impulse responseResersely when the impulse response of a system is known we can derive its difference equation. Following is an illustrative example.The impulse response of a system is periodic with the period of 3 indexes
h(n)=[1,2,3;1,2,3;1,2,3;1,2...] size 12{h \( n \) = \[ 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2 "." "." "." \] } {}
Find its input-output signal difference equation. Solution Delay the given impulse response 3 samplesh(n−3)=[0,0,0;1,2,3;1,2,3;1,2,3;1,2...] size 12{h \( n - 3 \) = \[ 0, matrix {
} 0, matrix {
} 0; matrix {
} 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2, matrix {
} 3; matrix {
} 1, matrix {
} 2 "." "." "." \] } {}Take the differenceh(n)−h(n−3)=[1,2,3;0,0,0,0,0...]=δ(n)+2δ(n−1)+3δ(n−2)alignl { stack {
size 12{h \( n \) - h \( n - 3 \) = \[ 1,2,3;0,0,0,0,0 "." "." "." \] } {} #
size 12{ matrix {
matrix {
matrix {
matrix {
matrix {
{} # {}
} {} # {}
} {} # {} # {}
} {} # {} # {}
} {} # {} # {}
} =δ \( n \) +2δ \( n - 1 \) +3δ \( n - 2 \) } {}
} } {}Then h(n)=h(n−3)+δ(n)+2δ(n−1)+3δ(n−2) size 12{h \( n \) =h \( n - 3 \) +δ \( n \) +2δ \( n - 1 \) +3δ \( n - 2 \) } {}By definition on of impulse response, the difference equation isy(n)=y(n−3)+x(n)+2x(n−1)+3x(n−2) size 12{y \( n \) =y \( n - 3 \) +x \( n \) +2x \( n - 1 \) +3x \( n - 2 \) } {}