10.2 Discrete-Time Fourier Transform 589
=π
[
δ(ω−ω 0 )e−jπ/^2 +δ(ω+ω 0 )ejπ/^2
]
=−jπ[δ(ω−ω 0 )−δ(ω+ω 0 )]
Thus, the frequency content of the cosine and the sine is concentrated at the frequencyω 0.
Although the sinusoids are infinite-energy signals they have finite power and their spectra can
be measured with a spectrum analyzer, which displays how the power is distributed over the
frequencies. n
10.2.7 Symmetry
When plotting or displaying the spectrum of a real-valued discrete-time signal it is important to know
that it is only necessary to show the magnitude and the phase spectra for frequencies [0π], since the
magnitude and the phase ofX(ejω)are even and odd functions ofω, respectively. This can be shown
by considering a real-valued discrete-time signalx[n], with inverse DTFT given by
x[n]=
1
2 π
∫π
−π
X(ejω)ejωndω
and its complex conjugate is
x∗[n]=
1
2 π
∫π
−π
X∗(ejω)e−jωndω=
1
2 π
∫π
−π
X∗(e−jω
′
)ejω
′n
dω′
Sincex[n]=x∗[n], asx[n] is real, comparing the above integrals we have that
X(ejω)=X∗(e−jω)
|X(ejω)|ejθ(ω)=|X(e−jω)|e−jθ(−ω)
Re[X(ejω)]+jIm[X(ejω)]=Re[X(e−jω)]−jIm[X(e−jω)]
or that the magnitude is an even function ofω—that is,
|X(ejω)|=|X(e−jω)| (10.20)
and that the phase is an odd function ofω, or
θ(ω)=−θ(−ω) (10.21)
Likewise, the real and the imaginary parts ofX(ejω)are also even and odd functions ofω:
Re[X(ejω)]=Re[X(e−jω)]
Im[X(ejω)]=−Im[X(e−jω)] (10.22)
