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Fundamentals of Multi-Rate Digital Signal Processing

Learn about the basics of sampling rate alteration devices: up-samplers, down-samplers, decimation, interpolation, and sampling rate conversion by rational factors in digital signal processing.

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Fundamentals of Multi-Rate Digital Signal Processing

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  1. Unit-V MULTI-RATE DIGITAL SIGNAL PROCESSING

  2. Multi-rate Digital Signal Processing Basic Sampling Rate Alteration Devices • Up-sampler - Used to increase the sampling rate by an integer factor • Down-sampler - Used to decrease the sampling rate by an integer factor

  3. DECIMATION

  4. DECIMATION

  5. y[n] x[n] M Down-Sampler Time-Domain Characterization • An down-sampler with a down-sampling factorM, where M is a positive integer, develops an output sequence y[n] with a sampling rate that is (1/M)-th of that of the input sequencex[n] • Block-diagram representation

  6. Down-Sampler • Down-sampling operation is implemented by keeping every M-th sample of x[n] and removing in-between samples to generate y[n] • Input-output relation y[n] = x[nM]

  7. D Decimation by a factor D In downsampling by an integer factor D>1, every D-th samples of the input sequence are kept and others are removed:

  8. Input sequence Output sequence Periodic train of impulses Decimation by a factor D • Relationship in time domain

  9. Decimation by a factor D • Relationship in frequency domain C

  10. C

  11. h(n) D Decimation by a factor D • Using a digital low-pass filter to prevent aliasing

  12. INTERPOLATION

  13. INTERPOLATION

  14. x[n] L Up-Sampler Time-Domain Characterization • An up-sampler with an up-sampling factorL, where L is a positive integer, develops an output sequence with a sampling rate that is L times larger than that of the input sequencex[n] • Block-diagram representation

  15. Up-Sampler • Up-sampling operation is implemented by inserting equidistant zero-valued samples between two consecutive samples ofx[n] • Input-output relation

  16. Up-Sampler • In practice, the zero-valued samples inserted by the up-sampler are replaced with appropriate nonzero values using some type of filtering process • Process is called interpolation and will be discussed later

  17. I h(n) Interpolation by a factor I In up-sampling by an integer factor I>1, I-1 equidistant zeros-valued samples are inserted between each two consecutive samples of the input sequence. Then a digital low-pass filter is applied.

  18. Input sequence Interpolation by a factor I Relationship in frequency domain

  19. I D Sampling period If is a rational number h1(n) h2(n) interpolation decimation Sampling rate conversion by a rational factor I/D

  20. I D h(n) Sampling Rate Conversion

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