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Communication Systems

Communication Systems. Prof. Chungming Kuo. Chapter 3. Fourier Transform Operations. Fourier Transform Operations. When signals are processed in the time domain, various changes occur in their spectra. It is important to understand how the spectra are affected by these operations.

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Communication Systems

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  1. Communication Systems Prof. Chungming Kuo

  2. Chapter 3 Fourier Transform Operations

  3. Fourier Transform Operations When signals are processed in the time domain, various changes occur in their spectra. It is important to understand how the spectra are affected by these operations.

  4. Fourier Transform Operations (cont.) In this module, the effects of various operations, such as differentiation and integration, will be studied in terms of the spectral changes.

  5. Fourier Transform Operation Pairs

  6. Effect of Differentiation on Spectrum

  7. Effect of Integration on Spectrum

  8. Delayed Function

  9. Effect of Modulation on Spectrum

  10. Effect of Time Scaling on Spectrum

  11. Spectral Convergence Time FunctionAmplitude Rolloff • Function has finite 1/f or discontinuity. -6 dB/octave • Slope has 1/f2 or finite discontinuity. -12 dB/octave • Neither function nor At least 1/f3 or slope has discontinuity. -18 dB/octave

  12. Summary • Any mathematical operation on a time function results in a change to the spectrum. • Differentiation accentuates high frequencies and diminishes low frequencies. • Integration accentuates low frequencies and diminishes high frequencies.

  13. Summary (cont.) • Multiplication by a complex exponential shifts the spectrum to higher frequencies. • “Speeding up” a signal results in a wider spectrum and vice-versa.

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