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Sec. 2–3 Power Spectral Density and Autocorrelation Function 63
w(t)
A
T
(a) Time Domain
1
f 0
|W(f)|
AT
––––
2
–f 0 0
1
–– T
1 ––T
t
f 0
f
(b) Frequency Domain (Magnitude Spectrum)
Figure 2–8 Waveform and spectrum of a switched sinusoid.
The spectrum for the switched sinusoid may also be evaluated using a convolution approach. The
multiplication theorem of Table 2–1 may be used where w 1 (t) =ß(t/T) and w 2 (t) = A sin v 0 t.
Here the spectrum of the switched sinusoid is obtained by working a convolution problem in the
frequency domain:
q
W (f) = W 1 (f) * W 2 (f) = W 1 (l) W 2 (f - l) dl
L
This convolution integral is easy to evaluate because the spectrum of w 2 (t) consists of two delta
functions. The details of this approach are left as a homework exercise for the reader.
-q
A summary of Fourier transform pairs is given in Table 2–2. Numerical techniques using
the discrete Fourier transform are discussed in Sec. 2–8.
2–3 POWER SPECTRALDENSITY AND AUTOCORRELATION
FUNCTION
Power Spectral Density
The normalized power of a waveform will now be related to its frequency domain description
by the use of a function called the power spectral density (PSD). The PSD is very useful in
describing how the power content of signals and noise is affected by filters and other devices