ABSTRACT
In this chapter, we introduce the concept of a system frequency response. First, we discuss the frequency response of continuous-time systems, while the discrete-time counterpart follows.
A system is completely specified by its impulse response h(t). The system response y(t) to an input signal x(t) is computed by convoluting the impulse response with the input signal; that is, the system response is given by
y(t) ¼ h(t) x(t): (8:1)
Applying Fourier transform to both sides of (8.1) yields
Y(V) ¼ H(V)X(V), (8:2)
where X(V) and Y(V) are the Fourier transforms of the input signal and of the output signal, respectively
H(V) is called frequency response, and is the Fourier transform of the impulse response h(t) of the system
The frequency response is a complete description of a system in the (cyclic) frequency domain. The mathematical expression of the frequency response is straightforwardly computed from Equation 8.2 as
H(V) ¼ F{h(t)} ¼ Y(V) X(V)
: (8:3)
The frequency response H(V) is usually a complex-valued function, and this is why sometimes it is denoted by H( jV). As any complex function, H(V) can be written in the form
H(V) ¼ jH(V)jejffH(V), (8:4)
where jH(V)j is the magnitude of H(V) ffH(V) is the phase of H(V)
h(t) ¼ e2tu(t). Plot the frequency response magnitude and phase for 0 V 10 rad=s. Finally verify Equation 8.4.
