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Chapter 2

Chapter 2 . Fourier Transform and Spectra Topics: Spectrum by Convolution Spectrum of a Switched Sinusoid Power Spectral Density Autocorrelation. Huseyin Bilgekul E eng3 60 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University.

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Chapter 2

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  1. 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:

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