ECE 6560 Multirate Signal Processing Analysis & Synthesis Notes

Transcription

1 Multirate Signal Processing Analysis & Synthesis Notes Dr. Bradley J. Bazuin Western Michigan University College of Engineering and Applied Sciences Departent of Electrical and Coputer Engineering 1903 W. Michigan Ave. Kalaazoo MI,

2 Harris Table of Contents 1. Why Multirate Filters? Thees of the aterial 2. The Resapling Process. Digital Filters & Windowing 3. Digital Filters. Fourier Transfor Fors and 4. Useful Classes of Filters. Applications 5. Systes That Use Resapling Filters. Parks-McClellan FIR Filter 6. Polyphase FIR Filters. Nyquist Filter 7. Resapling Filters. Filter-Deciation 8. Half-band Filters. Interpolation-Filter 9. Polyphase Channelizers. Polyphase Structures 10. Recursive Polyphase Filters. Analysis and Synthesis Banks 11. Cascade Integrator Cob Filters. Half-Band Structures 12. Cascade and Multiple Stage Filter Struct. CIC Structures 13. Counication Systes Applications. 2

3 Up and Down Sapling Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

4 Filter Deciation Band-liit using a low pass filter Eliinate the possibility of probles fro aliasing Reduce the saple rate using a deciator You ust pay attention to the Nyquist rate & aliasing Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

5 Interpolation Filter Interpolate by zero padding between saples Low Pass Filter to eliinate Spectral replicas An advanced alternative is to use a bandpass filter and keep only the desired replica! This work for deciation too! Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

6 Useful Perspectives for Multirate Filters A ultirate filter is not LTI (Linear Tie Invariant) A ultirate filter is a Linear Tie-Varying (LTV) process The ipulse response depends on which subfilter is connected to the output port when the input ipulse is presented to the filter. Since the output periodically revisits each coutator port, it is said that the ultirate filter is a Periodically Tie Varying (PTV) process. Note: this can ake it hard to copare inputs and outputs! You ust know which output you are coparing to or you could be generating an output between inputs which will never perfectly atch!? Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

7 Spectru Saple Rate Considerations for real signals Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

8 Filtering Needs Multirate processing involved: filter-deciation and interpolation-filter operations The definition of filters appropriate for the signals being processes is necessary for the signal processing being perfored. The desired or required passband bandwidths for the signals of interest ust be known. The filter structure is also an iportant consideration. Filter passbands, transition bands, stopbands, and other shape characteristics ust be defined and understood! Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

9 Generating Digital Filters For filters with inial passband ripple and narrow transition bands: 1. Generate an analog filter and then transfor it into a digital filter. (Not used in this textbook/class) 2. Use a windowed sinc function. (notes follow) Assues an ideal frequency band filter is convolved by a frequency doain window function. In the tie doain, the infinite sinc saples are truncated in length and shaped by a tie window. 3. Use a digital filter generating algorith Parks-McClellan is the ost popular (reez Algorith) Other iterative and non-iterative digital filter algoriths exist. Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

10 MATLAB Related functions kaiserord firpord fir1 firp Exaples Kaiser_FilterGen. PMcC_FilterGen. KaiserVsfirp. Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

11 Nyquist Filtering with Raised Cosine The Nyquist pulse is the wave shape required to counicate over band-liited channels with no ISI. It is generated as a raised cosine frequency spectru window Even syetric spectral window. Finite frequency width that is a fraction of the perfect reconstruction width. (i.e. /T) Preference to liit the tie response to a length 4T/ Truncated window (window length) and infinite sinc With convolution in the frequency doain, the spectru becoes a width of (1+)/T Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

12 Multirate Filtering Basics Filter-Deciation A filter is eployed to liit the signal bandwidth. The bandwidth and stopbands are designed so that the signal ay be appropriately deciated with little or no aliasing. Deciation reduces the signal saple rate to a desired rate. Interpolation-Filter An interpolator is used to increase the saple rate of the signal. Interpolation results in spectral replicas that ust be reoved by filtering so that only the desired signal band exists. The filter is eployed based on the required passband and stopband specification to eliinate unwanted replicas and keep the desired passband. Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

13 Polyphase Filter-Deciator Ipleentation y M 1 1 hr M x r 0 r0 M Coutate: separate the signal into M paths for low-pass filtering. Polyphase filter: the filters consist of all unique phases of the deciated filter! Su the result Coutator runs counter clockwise Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

14 Polyphase Interpolator-Filter y L 1 r0 s L h r xs r Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

15 15 Full Band Analysis Filter IFFT h 0 h 1 h 2 n x y 0 h M 1 y 1 y 2 y M 1 A Multichannel Digital Channelizer, pick any k exp r M k r x r h M k j y

16 Full Band Synthesis Filter x 0 h 0 x 1 h 1 yn x 2 h 2 x M 1 h M 1 y 1 sl t h r x s r r0 t N 1 k 0 k exp t k j2 N Each x is ixed to a different bin/band prior to polyphase interpolation. 16

17 FDM Generation and Processing Foring Wideband and Reforing Narrowband Filters narrower than the Nyquist regions are used for generating FDM wavefors. A guard band between adjacent frequencies is typically used x 0 h 0 h 0 y 0 x 1 h 1 h 1 y 1 yn xn x 2 FFT h 2 h 2 IFFT y 2 x M 1 h M 1 h M 1 y M 1 Synthesis OFDM Sybol Basis Analysis Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

18 Quadrature Mirror Filter Processing Observing Narrowband and Reforing Wideband Significant filter restriction are required if the output is required to approxiate the input! Quadrature Mirror Filter Definition and Requireents h 0 y 0 x 0 h 0 h 1 y 1 x 1 h 1 xn yn h 2 IFFT y 2 x 2 FFT h 2 h M 1 y M 1 x M 1 h M 1 Analysis Synthesis Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

19 Window Design of Half-Band Filter Chap8_4. Windowing provides additional side-lobe attenuation Kaiser windowing FirPM windowing h LP h LP h LP n 1 2 n sin 2 n 2 N : N 2 wn n 0.5*sinc.* kaiser2* N 1,5.8 ; N : N 2 n 0.5*sinc.* reez _ window2* N 1; Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN Passband narrows Ripple decreases Stopband iproves 19

20 Half-band Interpolation Interpolate Filter M= 2 Interpolate Filter Polyphase Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

21 Half-band Deciation Filter Deciate M=2 Deciating Polyphase Filter Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

22 Half-band Polyphase Deciation and Interpolation A Cascade of M=2 Polyphase Filters Can you build MATLAB code that does this? brute force versus polyphase filter tap lengths change due to bandwidths Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

23 Generalized Analysis Block Diagra Data Coutation with M block shifting. Polyphase filter foration of the atrix row eleents. Circular shift, odulo K, of the polyphase filter outputs. The IDFT to create the th saple for all K outputs. 23

24 Generalized Synthesis Block Diagra Data IDFT with extension (2x) as needed. Maintain ost recent results Modulo K selection of the polyphase filter row eleents. Polyphase row coputation and Coutated Data output. 24

25 Hogenauer Filter Using a CIC filter as the filter for interpolation-filter or filter-deciation Interpolation-Filter Filter-Deciation Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

26 Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN For one stage For two stages For K stages Multistage (Cascaded) IC Response z z z H M z z z H M K M z z z H sin 2 sin w wm e H jw 2 2 sin 2 sin w wm e H jw K jw w wm e H 2 sin 2 sin

27 FIR CIC clean-up Filters The sinc^m near-zero frequency response is often copensated for using an FIR filter H FIR e jw sin wm sin w 2 2 K Estiating useful narrowband bandwidth: Find attenuation required using 3 or 4 stage CIC Chap11_3. provides attenuation and alias curves WAG based on 3 stage single sideband (LPF), -60 db atten., approx. 1/10 th of the output saple rate (1/10*M) Notes and figures are based on or taken fro aterials in the course textbook: fredric j. harris, Multirate Signal Processing for Counication Systes, Prentice Hall PTR, ISBN

28 Suary Thees of the aterial Digital Filters & Windowing Fourier Transfor Fors and Applications Parks-McClellan FIR Filter Nyquist Filter Filter-Deciation Interpolation-Filter Polyphase Structures Analysis and Synthesis Banks Half-Band Structures CIC Structures 28