Lecture 11: Impulse Response of a Difference System, Characteristic
Polynomial and Stability
4.4 Impulse Response of a Difference LTI System
Consider the general form of a causal LTI difference system:
We briefly discuss one method to obtain the impulse response of this general causal LTI difference system initially at rest. It is similar to Method 1 for a continuous-time differential system, but it's simpler as there is no integration involved.
4.4.1 Impulse Response Obtained by Linear Combination of Impulse Responses of the Left-Hand Side of the Difference Equation
The step-by-step procedure to find the impulse response of a difference LTI system is as follows.
- Replace the whole right-hand side of the differential equation (4.46) by ,
- Find initial conditions on at times for a homogeneous response starting at time ,
- Calculate the homogeneous response to the homogeneous equation with these initial conditions,
- Finally, calculate as a linear combination of the current and delayed versions of .
Step 1: Under the assumption that the system is initially at rest, i.e.,
, let and replace the right-hand side of Equation (4.46) by a single unit impulse:
Step 2: To solve this equation for , we first observe that, by causality, the impulse can only be part of , not its delayed versions. This immediately gives us . Causality and the initial rest condition yield
Step 3: Thus starting at time , we need to solve the homogeneous equation
subject to the above initial conditions. Assume that the solution has the form of a complex exponential for . Substituting this exponential into Equation (4.48), we get a polynomial in "" multiplying an exponential on the left-hand side. Dividing both sides by and multiplying both sides by we get an equivalent equation:
and this equation holds if and only if the characteristic polynomial vanishes at the complex number :
By the fundamental theorem of algebra, this equation has at most N distinct roots. Assume that the N roots are distinct, and call them . This means that there are N distinct functions that satisfy the homogeneous equation (4.48). Then the solution to (4.48) can be written as a linear combination of these complex exponentials:
The complex coefficients can be computed using the initial conditions:
This set of linear equations can be written as
The Vandermonde matrix can be shown to be nonsingular. Hence a unique solution always exists for the 's which gives us the unique solution through Equation (4.51).
Step 4: Finally, by LTI properties of the difference system, the response of the left-hand side of (4.46) to its right-hand side is a linear combination of and delayed versions of it.
Therefore, the impulse response of the general causal LTI system described by the difference equation (4.46) is given by
4.5 Characteristic Polynomials and Stability of Differential and Difference Systems
4.5.1 The Characteristic Polynomial of an LTI Differential System
Recall that the characteristic polynomial of a causal differential LTI system of the type
is given by.(4.56)
This polynomial depends only on the coefficients on the left-hand side of the differential equation. It characterizes the intrinsic properties of the differential system as it doesn't depend on the input. The zeros of the characteristic polynomial are the exponents of the exponentials forming the homogeneous response, so they give us an indication of system properties, such as stability.
4.5.2 Stability of an LTI Differential System
We've seen that an LTI system is BIBO stable if and only if its impulse response is absolutely integrable. We have also figured out that a general formula for the impulse response of the system described by Equation (4.55) is given by
where the zeros of the characteristic polynomial are assumed to be distinct. Now we have to be concerned with the possibility of derivatives of impulses at in .
Recall that the first derivatives of are smooth or may have a finite jump (for ), while the impulse appears in . Thus, under the assumption that , the impulse response will have at worst a single impulse, which integrates to a finite value when integrated from to . Under these conditions, stability of the system is entirely determined by the exponentials in Equation (4.57):
We can see from the last upper bound above that the integral of the magnitude of the impulse response will be infinite if and only if is unbounded for some . This occurs if and only if . To summarize, with the above assumptions that the zeros of the characteristic polynomial are distinct and that , a causal LTI differential system is BIBO stable if and only if the real part of all of the zeros of its characteristic polynomial are negative (we say that they are in the left-half of the complex plane).
Example: Let's assess the stability of
The characteristic polynomial is which has its zero at . This system is therefore unstable, which is easy to see with an impulse response of the form (a growing exponential.)
4.5.3 The Characteristic Polynomial of an LTI Difference System
Recall that the characteristic polynomial of a causal difference LTI system of the type
is given by(4.61)
This polynomial depends only on the coefficients on the left-hand side of the difference equation. The zeros (assumed to be distinct) of the characteristic polynomial are the arguments of the exponentials forming the homogeneous response, so they also give us an indication of system properties, such as stability.
4.5.4 Stability of an LTI Difference System
Recall that an LTI difference system is stable if and only if its impulse response is absolutely summable. For the causal difference system above, this leads to the upper bound
and this last bound is finite if and only if for all . Hence the causal LTI difference system is BIBO stable if and only if all the zeros of its characteristic polynomial have a magnitude less than 1.
Example: Consider the causal first-order system
Its characteristic polynomial is , which has a single zero at . Hence this system is stable as .
4.6 Time Constant and Natural Frequency of a First-Order LTI Differential System
In general, the impulse response of an LTI differential system is a linear combination of complex exponentials of the type and their derivatives.
Consider the stable first-order system
Its impulse response is a single exponential: where and . The real number is called the natural frequency of the first-order system and its inverse is called the time constant of the first-order system. The time constant indicates the decay rate of the impulse response and the rise time of the step response. At time , the impulse response is , so the impulse response has decayed to 36.8% of its value at time .