Continuous-time LTI systems in the time domain
In the previous topic we focused on understanding discrete-time LTI systems. In this topic we briefly summarize the analogous results for continuous-time LTI systems. The main point is that if we replace the DT unit impulse with the CT unit impulse, and sums with integrals, essentially all of the ideas carry across directly. One practical difference is that computing convolutions in CT is more difficult involved than in DT, because it involves integration.
4.1 Continuous-time sifting formula
To see why this is true, make the change of variables s = t − τ in the integral. Then the RHS
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Continuous-time sifting formula
If x is a continuous-time signal that is continuous then ∞
x(τ)δ(t − τ) dτ for all times t.
where we have used the (defining) property of the CT unit impulse, that ∞
f (s)δ(s) ds = f (0) An alternative way to write the sifting formula is that any CT signal x that is continuous
the weights are exactly the numbers x(τ).
4.2 Impulse response of a continuous-time LTI system
The impulse response of a CT LTI system is defined to be the output of the system when the input is the unit impulse. The impulse response of a system is a signal. This signal is usually denoted h.
for any function continuous at 0. can be written as
x(t − s)δ(s) ds = x(t) −∞
This makes is clear that x is a weighted integral of time-shifted unit impulse functions, where
x(τ)(Delayτδ)(t) dτ.
Example 4.1
Consider the integrator system with input x and output y defined by t
The impulse response is h(t)=
the unit step function.
1 ift>0 δ(τ)dτ= 0 ift<0=u(t),
4.3 Impulse response and system properties
The impulse response characterizes a CT LTI system. Just as in the DT case, we can detect whether a CT LTI system is memoryless, causal, or BIBO stable, from the impulse response of the system. The characterisation are essentially identical to those given in the discrete-time setting, so we just state them here for reference.
Impulse response of memoryless LTI systems
A CT LTI system is memoryless if and only if its impulse response is a scaled copy of the unit impulse. In other words, the only memoryless LTI systems are gain systems.
Impulse response of causal LTI systems
A CT LTI system is causal if and only if its impulse response satisfies h(t) = 0 for all t < 0.
BIBO stability and impulse response of CT LTI systems
A CT LTI system is BIBO stable if and only if its impulse response h is absolutely inte- grable, i.e., there exists a positive real number M such that
−∞ |h(t)| dt ≤ M.
4.4 Continuous-time convolution and LTI system response
Just as in the DT case, the impulse response tells us everything about a CT LTI system. In particular, the same argument as in the DT case (but replacing sums with integrals, and replacing the DT unit impulse with the CT unit impulse signal), we can express the output of a CT LTI system in terms of the input and the impulse response.
If S is a CT LTI system with impulse response h, then y = S(x) if and only if
x(τ)h(t − τ) dτ. (4.1)
The integral (4.1) has special notation and terminology, analogous to the DT case.
We can now say, very concisely, how the input and output of a CT LTI system are re- lated.
This is exactly the same as for DT LTI systems, once we take into account the different definition of the unit impulse in CT, and the different definition of convolution in CT.
Just as in DT, the convolution operation satisfies • h∗δ=hforallsignalsh
• h1∗h2 =h2∗h1 forallsignalsh1 andh2
Similarly, the impulse response of series and parallel interconnections in CT are the same as in DT. If h1 and h2 are the impulse responses of two CT LTI systems then
• the impulse response of the parallel composition is h1 + h2 and • the impulse response of the serial composition is h1 ∗ h2
Continuous-time convolution
The convolution of CT signals x and h is a new CT signal h ∗ x defined by
(h ∗ x)(t) =
x(τ)h(t − τ) dτ. (4.2)
The output of a CT LTI system is the convolution of the input and the impulse response of the system.
Example 4.2
Let h(t) = u(t) − u(t − 1) be the impulse response of a CT LTI system. We will use the convolution formula to find the output y of the system if the input is x(t) = e−tu(t). Recall that u(t) is the unit step function. The impulse response h and input x are shown below:
The output y is the convolution of h and x:
Inside the integral, we should think of t as fixed and τ as the variable. At this point it is helpful to plot h(t − τ ) and x(τ ) (as functions of τ ) on the same set of axes. There are three cases to consider (based on the regions for which x and h are non-zero). The shaded area shows where both x(τ ) and h(t − τ ) are non-zero.
y(t) = (h ∗ x)(t) = =
x(τ)h(t − τ) dτ ∞ −τ
h(t−τ) x(τ) 1
t−1t0 x(τ )
t−10t x(τ )
e h(t − τ ) dτ
Note that h(t−τ) = 1 if t−1 < τ ≤ t and h(t−τ) = 0 otherwise. We can simplify the
integral for y to obtain
Since b e−τ dτ = e−a − e−b we conclude that a
y(t) = 1 − e−t e−t+1 − e−t
if 0 < t ≤ 1 if t > 1.
0 if t ≤ 0 y(t)=te−τ dτ if0
4.5 Summary
In this topic we determined the output of a continuous-time LTI system given the input. The basic approach parallels the discrete-time case but with sums replaced with integrals, and the DT unit impulse replaced with the CT unit impulse. Any CT signal can be written as an integral of scaled, time-shifted unit impulse signals. Using linearity and time-invariance, it is enough to know the output of the system when the input is the unit impulse (the impulse response). The output to a general input is then an integral of scaled, time-shifted, copies of the impulse response. The convolution integral conveniently packages this argument into a formula.
The impulse responses of certain simple CT systems (such as gain and delays and integrators) are fairly easy to determine. Just as in discrete-time, the impulse response of parallel and serial compositions can be found explicitly in terms of the impulse responses of the systems being interconnected.
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