Linear Systems: Discrete-Time Impulse Response Descriptions

Abstract

An important input-output description of a linear discrete-time system is its (discrete-time) impulse response (or pulse response), which is the response h(k, k₀) to a discrete impulse applied at time k₀. In time-invariant systems that are also causal and at rest at time zero, the impulse response is h(k, 0), and its z-transform is the transfer function of the system. Expressions for h(k, k₀) when the system is described by state variable equations are derived.

Introduction

Consider linear, discrete-time dynamical systems that can be described by

$$\displaystyle{ y(k) =\sum \limits _{ l=-\infty }^{+\infty }H(k,l)u(l) }$$

(1)

where $k,l \in \mathbb{Z}$ is the set of integers, the output is $y(k) \in \mathbb{R}^{p}$, the input is $u(k) \in \mathbb{R}^{m}$, and $H(k,l) : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{R}^{p\times m}$. For instance, any system that can be written in state variable form

$$\displaystyle{ \begin{array}{l} x(k + 1) = A(k)x(k) + B(k)u(k) \\ \quad y(k) = C(k)x(k) + D(k)u(k)\\ \end{array} }$$

(2)

or

$$\displaystyle{ \begin{array}{l} x(k + 1) = Ax(k) + Bu(k) \\ \quad y(k) = Cx(k) + Du(k)\\ \end{array} }$$

(3)

can be represented by (1). Note that it is assumed that at $l = -\infty$, the system is at rest, i.e., no energy is stored in the system at time −∞.

Define the discrete-time impulse (or unit pulse) as

$$\displaystyle{\delta (k) = \left \{\begin{array}{*{20}{c}} 1&\quad k = 0& \\ 0 & \quad k\neq 0 &, k \in \mathbb{Z} \end{array} \right.}$$

and consider a single-input, single-output system:

$$\displaystyle{ y(k) =\sum \limits _{ l=-\infty }^{+\infty }h(k,l)u(l) }$$

(4)

If $u(l) =\delta (\hat{l} - l)$, that is, the input is a unit pulse applied at $l =\hat{ l}$, then the output is

$$\displaystyle{y_{I}(k) = h(k,\hat{l}),}$$

i.e., $h(k,\hat{l})$ is the output at time k when a unit pulse is applied at time $\hat{l}$.

So in (4) h(k, l) is the response at time k to a discrete-time impulse (unit pulse) applied at time l. h(k, l) is the discrete-time impulse response of the system. Clearly if h(k, l) is known, the response of the system to any input can be determined via (4). So h(k, l) is an input/output description of the system.

Equation (1) is a generalization of (4) for the multi-input, multi-output case. If we let all the components of u(l) in (1) be zero except for the jth component, then

$$\displaystyle{ y_{i}(k) =\sum \limits _{ l=-\infty }^{+\infty }h_{ ij}(k,l)u_{j}(l) }$$

(5)

h _ij (k, l) denotes the response of the ith component of the output of system (1) at time k due to a discrete impulse applied to the jth component of the input at time l with all remaining components of the input being zero. H(k, l) = [h _ij (k, l)] is called the impulse response matrix of the system.

If it is known that system (1) is causal, then the output will be zero before an input is applied. Therefore,

$$\displaystyle{ H(k,l) = 0,\quad \text{ for }k <l, }$$

(6)

and so when causality is present, (1) becomes

$$\displaystyle{ y(k) =\sum \limits _{ l=-\infty }^{k}H(k,l)u(l). }$$

(7)

A system described by (1) is at rest at k = k₀ if u(k) = 0 for $k\geqslant k_{0}$ implies y(k) = 0 for $k\geqslant k_{0}$. For a system at rest at k = k₀, (7) becomes

$$\displaystyle{ y(k) =\sum \limits _{ l=k_{0}}^{k}H(k,l)u(l). }$$

(8)

If system (1) is time-invariant, then $H(k,l) = H(k - l,0)$ (also written as H(k − l)) since only the time elapsed (k − l) from the application of the discrete-time impulse is important. Then (8) becomes

$$\displaystyle{ y(k) =\sum \limits _{ l=0}^{k}H(k - l)u(l),\quad k \geq 0, }$$

(9)

where we chose k₀ = 0 without loss of generality. Equation (9) is the description for casual, time-invariant systems, at rest at k = 0.

Equation (9) is a convolution sum and if we take the (one-sided or unilateral) z-transform of both sides,

$$\displaystyle{ \hat{y}(z) =\hat{ H}(z)\hat{u}(z), }$$

(10)

where $\hat{y}(z)$, $\hat{u}(z)$ are the z-transforms of y(k), u(k) and $\hat{H}(z)$ is the z-transform of the discrete-time impulse response H(k). $\hat{H}(z)$ is the transfer function matrix of the system. Note that the transfer function of a linear, time-invariant system is defined as the rational matrix $\hat{H}(z)$ that satisfies (10) for any input and its corresponding output assuming zero initial conditions.

Connections to State Variable Descriptions

When a system is described by (2), then

$$\displaystyle\begin{array}{rcl} y(k)& =& \sum \limits _{l=k_{0}}^{k-1}C(k)\Phi (k,l + 1)B(l)u(l) \\ & & +D(k)u(k),\quad k> k_{0} {}\end{array}$$

(11)

where it was assumed that x(k₀) = 0, i.e., the system is at rest at k₀. Here $\Phi (k,l)$ ($= A(k - 1)\cdots A(l)$) is the state transition matrix of the system.

Comparing (11) with (8), the discrete-time impulse response of the system is

$$\displaystyle{ H(k,l) = \left \{\begin{array}{*{20}{c}} C(k)\Phi (k,l + 1)B(l)&\quad k> l \\ D(k) &\quad k = l \\ 0 &\quad k <l \end{array} \right. }$$

(12)

Similarly, when the system is time-invariant and is described by (3),

$$\displaystyle{ y(k) =\sum \limits _{ l=k_{0}}^{k-1}CA^{k-(l+1)}Bu(l) + Du(k),\ \ k> k_{ 0} }$$

(13)

where x(k₀) = 0 and

$$\displaystyle{ H(k,l) = H(k-l) = \left \{\begin{array}{*{20}{c}} CA^{k-(l+1)}B &\quad k> l \\ D &\quad k = l\\ 0 &\quad k <l \end{array} \right. }$$

(14)

When l = 0 (taking the time when the discrete impulse is applied to be zero, l = 0), the discrete-time impulse response is

$$\displaystyle{ H(k) = \left \{\begin{array}{*{20}{c}} CA^{k-1}B &\quad k> 0 \\ D &\quad k = 0\\ 0 &\quad k <0 \end{array} \right. }$$

(15)

Taking (one-sided or unilateral) z-transforms of both sides in (15),

$$\displaystyle{ \hat{H}(z) = C(zI - A)^{-1}B + D }$$

(16)

which is the transfer function matrix in terms of the coefficient matrices in the state variable description (3). Note that (16) can also be derived directly from (3) by assuming zero initial conditions (x(0) = 0) and taking z-transforms of both sides.

Finally, it is easy to show that equivalent state variable descriptions give rise to the same discrete-impulse response.

Summary

The discrete-time impulse response is an external, input-output description of linear, discrete-time systems. When the system is time-invariant, the z-transform of the impulse response h(k, 0) (which is the output response at time k due to a discrete impulse applied at time zero with initial conditions taken to be zero) is the transfer function – another very common input-output description. The relationships to the state variable descriptions were shown.

Cross-References

• Linear Systems: Discrete-Time, Time-Invariant State Variable Descriptions

• Linear Systems: Discrete-Time, Time-Varying, State Variable Descriptions

Recommended Reading

External or input-output descriptions such as the impulse response and the transfer function (in the time-invariant case) are described in several textbooks below.