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Class Report

Class Report. 馬睿峰 : Multirate Signal Processing Introduction. Multirate Signal Processing. Sampling rate conversion : The process of converting a signal from a given rate to a different rate Multirate Systems : Systems handling more than one sampling rate

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Class Report

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  1. Class Report 馬睿峰 : Multirate Signal Processing Introduction

  2. Multirate Signal Processing • Sampling rate conversion: The process of converting a signal from a given rate to a different rate • Multirate Systems: Systems handling more than one sampling rate • Objective: Minimize the complexity of the signal processing under Nyquist Sampling criterion • Key Operations: Sampling rate conversion for communications -- Decimation: Down sampling at receiver side -- Interpolation: Up sampling at transmission side • DSP Functions: Decimation Filtering Interpolation Filtering

  3. Why and Where to Use Multirate • Multirate systems can reduce required system computation, bandwidth, or storage requirements • Multirate systems are used for converting data between A/D and D/A devices that run at different sampling rates • Common multirate applications: • Sampling systems • Audio/speech encoders • Image/video encoders • Music synthesizers • CAT scan, image systems • Echo cancellers • Modems, data multiplexers

  4. Sampling Rate Conversion • It is often advantages to digitally alter the sampling rate during processing, after the initial A/D conversion has been finished: • High sampling rates (perhaps many times the Nyquist rates) can be used to help remove noise and quantization error • Low sampling rates (near the Nyquist rate) are best for computation speed and memory requirements

  5. Sampling – Time Domain (Multiplication)  Analog to digital conversion process: s(t) xc(t) x[n] xc(t) s(t) x[n] = xc(nt)

  6. Sampling – Frequency Domain (Convolution)  If a continuous signal xc(t) is band limited with |Xc(j)| = 0, for ||  2Fc, then xc(t) can be uniquely reconstructed without error, if Fsam  2Fc, where Fsam= 1/T is the sampling frequency. Xc(j) -2Fc 0 2Fc  S(j) -2F -F 0 F=2F  Xc(j)  S(j) 0 

  7. Aliasing (Under Sampling)  Fourier transform of continuous-time signal -/T /T  Fourier transform of sampled signal -   2 -2 High Frequency noise

  8. Multirate Antialiasing Filter Analog LPF Digital LPF 3 A/D 4

  9. Sampling and Reconstruction Sampling Reconstruction Nyquist sampled Under sampled Nyquist Condition:

  10. Multirate signal processing - Decimation The M-fold decimator : The M-fold decimator which takes an input sequence x(n) and produces the output sequence y[n] = x[Mn] where M is an integer. Only those samples of x[n] which occur at time equal to multiples of M are retained by the decimator . x[n] w[n] y[m] h[n] M F F F’ Over-sampled signal Decimated signal 3 to 1 decimation time time 0 0

  11. Decimation • The decimator is also called as downsampler , subsampler, sampling ratecompressoror merely acompressor • Aliasing can be avoided if x[n] is a lowpass signal bandlimited to the region |  | < /M • In most applications ,the decimator is preceeded by a lowpass filter called the decimation filter . The filter ensures that signal being decimated is bandlimited

  12. Decimation by Integer Factor x[n] w[n] y[m] h[n] M F F F’ Time-domain input-output relation: Frequency-domain input-output relation: Idea frequency response of decimation filter:

  13. Decimation by Integer Factor x[n] w[n] y[m] h[n] M F F F’ |X(ej)| F 0  2 |H(ej)| F 0 /M  2 |W(ej)| F 0 /M  2 |Y(ej’)| F’=F/M 0  2 4 6

  14. Interpolation by Integer Factor Definition: A up-sampler with an up-sampling factor L. The aim is to get a new sequence with with a higher sampling frequency without changing the spectrum. y[m] w[m] x[n] L h[m] F’ F F’ 1 to 3 interpolation time time             0 0

  15. Interpolation by Integer Factor y[m] w[m] x[n] L h[m] F’ = L  F F F’ Time-domain input-output relation: Frequency-domain input-output relation: Idea frequency response of Interpolation filter:

  16. Interpolation by Integer Factor y[m] w[m] x[n] L h[m] F’ = L  F F F’ |X(ej)| 0  2 |W(ej’)| F’ /L  0 2 |Y(ej’)| F’ 0 /L  2

  17. Conversion by Rational Factor M/L Interpolation by L Decimation by M x[n] y(m) s[k] h1(k) L h2(k) M F F’ = (L/M)F F”=LF x[n] y[m] h(k) L M F F”=LF F” F’ = (L/M)F

  18. References of Multirate Signal Processing • Ronald E. Crochiere and Lawrence R. Rabiner, Multirate Digital Signal Processing, Prentice Hall, 1983. • P. P. Vaidyanathan, Multirate Systems and Filter Bank, Prentice Hall, 1993. • P. P. Vaidyanathan, Multirate Digital Filters, Filter Banks, Polyphase Networks and Applications: A Tutorial, IEEE Proceeding, Vol.78, Jan. 1990.

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