Signals and Systems - Electrical Engineering

5.6 Spectral Representation 323

since the integral can be thought of as an infinite sum of complex values. Comparing the two
integrals, we have that

X()=X∗(−)

or

|X()|ejθ()=|X(−)|e−jθ(−)

whereθ()=∠(X())is the phase ofX(). We can then see that

|X()|=|X(−)|

θ()=−θ(−)

or that the magnitude is an even function ofand the phase is an odd function of. It can also be
seen that

Re[X()]=Re[X(−)]

Im[X()]=−Im[X(−)]

or that the real part of the Fourier transform is an even function and that the imaginary part of the
Fourier transform is an odd function of.

Remarks

n Clearly, if the signal is complex, the above symmetry will not hold. For instance, if x(t)=ej^0 t=
cos( 0 t)+jsin( 0 t), using the frequency-shift property its Fourier transform is
X()= 2 πδ(−o)

which occurs at= 0 only, so the symmetry in the magnitude and phase does not exist.
n It is important to recognize the meaning of “negative” frequencies. In reality, only positive frequencies
exist and can be measured, but as shown the spectrum, magnitude or phase, of a real-valued signal
requires negative frequencies. It is only under this context that negative frequencies should be understood
as necessary to generate “real-valued” signals.

nExample 5.11

Use MATLAB to compute the Fourier transform of the following signals:

(a)x 1 (t)=u(t)−u(t− 1 )

(b)x 2 (t)=e−tu(t)

Plot their magnitude and phase spectra.

Solution

Three possible ways to compute the Fourier transforms of these signals using MATLAB are: (1)

find their Laplace transforms as in Chapter 3 usinglaplaceand compute the magnitude and phase