Impulse Response and its Properties of Various LTI Systems.
For discrete time the representation takes the form of the convolution sum, while
it’s continuous time counterpart is the convolution integral.
For Discrete time:
+∞
y[n]= x[n]*h[n] = ∑ x[k] h[n-k]......................(1)
k=-∞
For Continuous time:
+∞
y(t)=∫-∞ x(τ) h(t-τ) dτ.........................................(2)
From above expressions we can conclude that the characteristics of an LTI system are
completely determined by its impulse response.
1. The Commutative Property:
A basic property of the convolution in both continuous and discrete time is that it
is a commutative operation.
In discrete time:
+∞
x[n]*h[n]=h[n]*x[n]= ∑ h[n] x[n-k]
k=-∞
In continuous Time:
+∞
x(t)*h(t)=h(t)*x(t)= ∫-∞ h(τ) x(t-τ)dτ
From the above expressions, the output of an LTI system with input x[n] and unit
impulse response h[n] is identical to the output of an LTI system with input h[n] and unit
impulse response x[n]. Similarly, the output of an LTI system with input x(t) and unit
impulse response h(t) is identical to the output of an LTI system with input h(t)and unit
impulse response x(t).
2. The Distributive Property:
In discrete time:
x[n]*(h1[n]+h2[n])= x[n]*h1[n]+x[n]*h2[n]
In continuous time:
X(t)*[h1(t)+h2(t)]=x(t)*h1(t)+x(t)*h2(t)
The distributive property has a useful interpretation in terms of system
interconnections. Consider two continuous time LTI systems in parallel a shown in
figure. The systems shown in the block diagram are LTI system with the indicated unit
impulse responses.
h1(t)
x(t)
y(t)
x(t)
h1(t)+h2(t)
h2(t)
(a)
(b)
y(t)
The two systems with impulse responses h1(t) and h2(t), have identical inputs and
their outputs are added. So,
y1(t) =x(t)*h1(t)
And
y2(t) =x(t)*h2(t),
The system of fig (a) has output,
y(t) =x(t)*h1(t)+x(t)*h2(t),
And the system of figure (b) has output,
y(t) =x(t)*[h1(t)+h2(t)]
Hence, by the virtue of the distributive property of convolution, a parallel
combination of LTI systems can be replaced by single LTI system whose unit impulse
response is the sum of the individual unit impulse responses in the parallel combination.
3. The Associative Property:
It is an important and useful property of convolution.
In discrete time;
x[n]*(h1[n]*h2[n])=(x[n]*h1[n])*h2[n]
In continuous time
x(t)*[h1(t)*h2(t)]=[x(t)*h1(t)]*h2(t)
From the block diagram shown, an interpretation of the associative property can
be illustrated. In the Figure (a)
y[n]=w[n]*h2[n]
=(x[n]*h1[n])*h2[n]
In the figure (b)
y[n]=x[n]*h[n]
=x[n]*(h1[n]*h2[n])
x[n]
h1[n]
h2[n]
y[n]
w[n]
(a)
x[n]
h[n]=h1[n]+h2[n]
(b)
y[n]
According to the associative property, the series interconnection of the two
systems in fig(a) is equivalent to the single system in fig(b).This can be generalized to an
arbitrary number of LTI systems in cascade. This interpretation and conclusion also hold
in continuous time. From Figures we can conclude that the impulse response of the
cascade of two LTI systems is the convolution of their individual impulse responses.
The unit impulse response of a cascade of two LTI systems does not depend on the order
in which they are cascaded. The order in which they are cascaded does not matter as far
as the overall system impulse response is concerned.
LTI systems with or without memory:
A system is memoryless if its output at any time depends only on the value of the
input at the same time.
For discrete time LTI system, if h[n] =0 for n#0.in this case the impulse response
has the form
h[n]=Kδ[n],
Where K=h[0] is a constant and the convolution sum reduces to the relation
y[n]=Kx[n]
If a discrete time LTI system has an impulse response h[n] that is not identically
zero for n#0, then system has memory.
A continuous time LTI system is memoryless if h(t)=0 fot t#0 and such a
memoryless LTI system has the form
y(t)=Kx(t)
For some constant K and has the impulse response
h(t)=Kδ(t)
Causality for LTI systems
A system is said to be causal if the output at any time depends on values of the
input at the present time and in the past. If two inputs to a casual system are identical up
to some point in time to or no, the corresponding outputs must also be equal up to this
same time.
In order for the discrete time to LTI systems to be casual, y[n] must not depend on
x[k] for k>n. We have,
From above equation, for this to be true, all the coefficients h[n-k] that multiply
values of x[k] for k>n must be zero. Then the impulse response of a casual discrete time
LTI system satisfy the condition
h[n] =0 for n<0........................(i)
Hence, the impulse response of a casual LTI system must be zero before the impulse
occurs.
For casual discrete time LTI system, from condition (i), the convolution sum
representation in eq (1) becomes
n
y[n]= ∑ x[n] h[n-k],
k=-∞
Similarly, for continuous time LTI system, the required condition for system to be casual
is,
h(t)=0, t<0
and the convolution integral is given by,
t
-∞
y(t)= ∫-∞ x(τ)h(t-τ) dτ= ∫0 h(τ)x(t-τ) dτ
Stability of the LTI systems:
A system is stable if every bounded input produces a bounded output.
In order to determine the conditions under which LTI systems are stable.
Let an input x(t) that is bounded in magnitude:
x(t)<B, for all t..............................(i)
Let LTI system is apply with unit impulse response h (t), then using convolution sum, we
obtain an expression for the magnitude of the output.i.e
+∞
│y(t)│=│ ∫-∞ h(τ)x(t-τ) dτ│.............(ii)
Since the maginitude of the sum of a set of numbers in no larger than the sum of
the magnitudes of the numbers, hence,
+∞
│y(t)│≤ ∫-∞ │ h(τ)││x(t-τ) │dτ................(iii)
From eq.(i),│x(t-τ)│<B for all values of t and τ.also from eq.(iii), we get,
+∞
│y(t)│ ≤ B∫-∞│h(τ)│dτ...........................(iv)
Hence the system is stable if the impulse response is absolutely integrable, i.e., if
+∞
∫-∞│h(τ)│dτ <∞
Similarly, for discrete time system,
│x[n]│<B for all n
Then in analogy with eq. (i)-(iv), we have,
+∞
│y[n]│=│∑ h[k] x[n-k]│
k=-∞
+∞
≤ ∑│h[k] ││x[n-k]│
K=-∞
+∞
≤ B ∑│h[k] │
K=-∞
Therefore the system is stable if the impulse response is absolutely summable, i.e. if
+∞
∑ │h[k] │ < ∞
K=-∞
Invertibilty of LTI system:
A system is invertible only if an inverse system exists when connected in series
with the original system and produces an output equal to the input to the first system.
Consequently, if LTI system is invertible then it has an LTI inverse.
Let us consider a continuous time system shown in the figure below with impulse
response h1(t) which results in x(t) such that series interconnection in figure(a) is
identical to the identity system in figure(b).
x(t)
h(t)
h1 (t)
w(t)=y(t)
y(t)
(a)
Identity system δ(t)
x(t)
x(t)
(b)
Hence, for continuous time LTI system, the impulse response h1(t) of the inverse system
for an LTI system with impulse response h(t) if,
h(t)*h1(t)=δ(t)
Similarly, for discrete time system,
h[n]*h1[n]=δ[n].