Signals and Systems - Electrical Engineering

4.2 Eigenfunctions Revisited 239

where we letH(j 0 )equal the integral in the second equation. The input signal appears in the output
modified by the frequency response of the systemH(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 timet—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 timet, 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 signalx(t)is a linear combination of complex exponentials, with different
amplitudes, frequencies, and phases, or

x(t)=

∑

k

Xkejkt

where Xk are complex values, since the output corresponding toXkejkt is XkejktH(jk) by
superposition the response tox(t)is

y(t)=

∑

k

XkejktH(jk)

=

∑

k

Xk|H(jk)|ej(kt+∠H(jk)) (4.3)

The above is valid for any signal that is a combination of exponentials of arbitrary frequencies. As we
will see in this chapter, whenx(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

x(t)=

1

2 π

∫∞

−∞

X()ejtd

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

y(t)=

1

2 π

∫∞

−∞

X()ejtH(j)d

=

1

2 π

∫∞

−∞

X()|H(j)|e(jt+j∠H(j))d (4.4)