PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 337

R.4.57 Let the input to a system be x(t) = δ(t), since X(w) = ℑ{δ(t)} = 1, then

Y(w) = H(w)

where h(t) ↔ H(w).

Recall that h(t) is referred to as the impulse system response, for obvious reasons.

R.4.58 Recall some useful convolution properties (from Chapter 1)

a. f 1 (t) ⊗ f 2 (t) = f 2 (t) ⊗ f 1 (t)

b. f 1 (t) ⊗ [f 2 (t) + f 3 (t)] = f 1 (t) ⊗ f 2 (t) + f 1 (t) ⊗ f 3 (t)

R.4.59 As a consequence of the convolution properties of R.4.58, the block box system

transformation diagrams shown in Figure 4.5 can be obtained.

Note that the system input and the system equation can be interchanged with-

out affecting its output. Also note that there is no distinction between a signal or

system, and the analytical tools developed for signals can be applied equally well

for systems.

R.4.60 Recall that a system can be thought of as a method or algorithm of processing or

changing an input signal x(t) (or x(n)) i nto an out put sig nal y(t) (or y(n)). If the system

is linear then it can be characterized by h(t) or H(w), its system impulse response in

time, or its FT in frequency. The system transfer function H(w) thus modifi es, or fi l-

ters the spectrum of the input X(w). The objective of the system transfer function is

to change the relative importance of the frequencies contained in the input signal,

both in amplitude and phase. H(w) is also called the gain function since it weights

the various frequencies’ components of the input x(t) to generate its output y(t).

R.4.61 Let x(t) be the input to a linear system, then its output y(t) is said to be distortionless

if it is of the form y(t) = kx(t − a), where k and a are constants, referred as the gain

and delay, respectively.

R.4.62 The distortionless time–frequency relation is then given by

kx t a KX w e

() → ()jwa

See R.4.48, property d. Therefore, for distortionless transmission the system

returns a constant gain K, and a linear phase shift of the form −aw, when its input

h(t) y(t) x(t)⊗= h(t)

x(t)

FIGURE 4.3

System expressed in the time domain.

X(w) H(w) Y(w) = H(w)X(w)

FIGURE 4.4

System expressed in the frequency domain.