EEE 431 1

1

• Fourier Transform and Spectra
• Topics
• Spectrum by Convolution
• Spectrum of a Switched Sinusoid
• Power Spectral Density
• Autocorrelation

2
Spectrum of a triangular pulse by convolution
The tails of the triangular pulse decay faster
than the rectangular pulse. WHY ??
3
Spectrum of a Switched Sinusoid
Using the Frequency Translation Property of the
Fourier Transform
We can get a similar result using the convolution
property of the Fourier Transform.
4
Spectrum of a Switched Sinusoid
5
Power Spectral Density (PSD)

• We define the truncated version (Windowed) of the
  waveform by

• The average normalized power from the time
  domain

• Using Parsevals theorem to calculate power
  from the frequency domain

6
Power Spectral Density

• Definition The Power Spectral Density (PSD) for
  a deterministic power waveform is

• where wT(t) ? WT(f) and Pw(f) has units of
  watts per hertz.
• The PSD is always a real nonnegative function of
  frequency.
• PSD is not sensitive to the phase spectrum of
  w(t)
• The normalized average power is
• This means the area under the PSD function is
  the normalized average power.

7
Autocorrelation Function

• Definition The autocorrelation of a real
  (physical) waveform is

• Wiener-Khintchine Theorem PSD and the
  autocorrelation function are Fourier transform
  pairs

• The PSD can be evaluated by either of the
  following two methods
• Direct method by using the definition,
• Indirect method by first evaluating the
  autocorrelation function and then taking the
  Fourier transform
• Pw(f) I Rw(t)

• The average power can be obtained by any of the
  four techniques.

8
PSD of a Sinusoid
9
PSD of a Sinusoid

• The average normalized power may be obtained by
  using