1. Fourier Transform and Spectra
Chapter 2
Fourier Transform and Spectra
Topics:
Spectrum by Convolution
Spectrum of a Switched Sinusoid
Power Spectral Density
Autocorrelation
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

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 Parseval’s 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(τ) ]
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: