Eigenfunctions Revisited 239

4.2 Eigenfunctions Revisited 239

where we let H( j 0 ) equal the integral in the second equation. The input signal appears in the output modified by the frequency response of the system H( j 0 ) at the frequency  0 of the input. Notice that the convolution integral limits indicate that the input started at −∞ and that we are considering the output at finite time t—this means that we are in steady state. The steady-state response of a stable LTI system is attained by either considering that the initial time when the input is applied to the system is −∞ and we reach a finite time t, or by starting at time 0 and going to ∞.

The above result for one frequency can be easily extended to the case of several frequencies present at the input. If the input signal x(t) is a linear combination of complex exponentials, with different amplitudes, frequencies, and phases, or

x(t) X X e j k = t

kk

where X are complex values, since the output corresponding to X e j k t is X e j k k t k k H( j k ) by superposition the response to x(t) is

y(t) X X e j k = t

H( j k )

X ) j( k = t X

|H( j )) k |e +∠H( j k

The above is valid for any signal that is a combination of exponentials of arbitrary frequencies. As we will see in this chapter, when x(t) is periodic it can be represented by the Fourier series, which is a combination of complex exponentials harmonically related (i.e., the frequencies of the exponentials are multiples of the fundamental frequency of the periodic signal). Thus, when a periodic signal is applied to a causal and stable LTI system its output is computed as in Equation (4.3).

The significance of the eigenfunction property is also seen when the input signal is an integral (a sum, after all) of complex exponentials, with continuously varying frequency, as the integrand. That is, if

Z 1 ∞ x(t) jt = X()e d

then using superposition and the eigenfunction property of a stable LTI system, with frequency response H( j), the output is

1 jt

y(t) =

X()e H( j)d

( = jt X() |H( j)|e +j∠H( j)) d

C H A P T E R 4: Frequency Analysis: The Fourier Series

The above representation of x(t) corresponds to the Fourier representation of aperiodic signals, which will be covered in Chapter 5. Again here, the eigenfunction property of LTI systems provides an effi- cient way to compute the output. Furthermore, we also find that by letting Y() = X()H( j) the above equation gives an expression to compute y(t) from Y(). The product Y() = X()H( j) cor- responds to the Fourier transform of the convolution integral y(t) = x(t) ∗ h(t), and is connected with the convolution property of the Laplace transform. It is important to start noticing these connections, to understand the link between Laplace and Fourier analysis.