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 & Erhan A. Ince Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

Spectrum of a triangular pulse by convolution The tails of the triangular pulse decay faster than the rectangular pulse. WHY ??

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.

Spectrum of a Switched Sinusoid

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

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.

       

 

Normalized power for a periodic waveform   where cn are the complex Fourier coefficients of the waveform

 

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.

PSD of a Sinusoid

PSD of a Sinusoid The average normalized power may be obtained by using: