ECE 6560 Multirate Signal Processing Lecture 9

Transcription

1 Multirate Signal Processing Lecture 9 Dr. Bradley J. Bazuin estern Michigan University College of Engineering and Applied Sciences Department of Electrical and Computer Engineering 193. Michigan Ave. Kalamazoo MI,

2 Chapter 9: Polyphase Channelizers 9.1 Demodulator Channel Ban Arbitrary Output Sample Rates Comparison of Design Options 24 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

3 Digital Filter Ban Channelizer Demodulation of an entire set of channels simultaneously Downconvert, filter, and decimate each of the channel bands. Applications: Multichannel radios Spectrum Analyzer otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

4 Polyphase FIR Review otes from Chapter 6 A channelizer is based on performing multiple bandpass-filter polyphase FIR structures simultaneously! The band-pass mixing (either from a center frequency to baseband or of the filter from baseband to the desired frequency) periodicity is related to he number of polyphase filter elements. This relationship allows the use of a discreet Fourier transform (with all the related theories) to generate simultaneous outputs. The DFT size is the same as the number of polyphase filter elements! otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

5 Channelizer otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

6 Figure 6.3 Spectra Spectra observed at various points in the processing chain of a standard radio receiver Down Converter otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

7 hat about Mixing Prior to Filter Decimation xn hn wn M y m n M y m hn xmm n n exp j2 mm n y m hn xmm nexp j2 m exp j2 n y Let M M m hn xmm nexp j2 n otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB n n 7

8 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB Mixing Continued Let n rm M rm j rm mm x rm h m y M r 2 exp 1 1 r m x M j r h m y M r 2 exp exp r M r m x r h M j m y The complex weighting and summation of M -tap filters. The computation is performed once every M input samples.

9 Simple Analysis Systems The previous analysis was for a simple critically sampled case with M=K=. May not be typical of desired communication system processing. ADC/input rates based on comm. bands and available devices. Magic numbers used to define K and bin spacings. Analysis output rates often based on data symbol or other embedded data signaling rates. Therefore, we need to allow integer oversampling and rational fraction oversampling. K/M may be a rational fraction not an integer! otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

10 Filter-Decimation Reformed ote: mixing uses K where X m l hl mm l xmml K (1) For a filter of length where K n exp i2n K (2) hn n hn n n (3) K, l r (4) X 1 m m hr M r xmm r r1 (5) otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

11 Filter-Decimation (Polyphase-FFT) xn hn M X m n K Let (6) Cases: X (7) r 1 1 m m hr xmm r M 1) Critically sampled, M 2) Integer ratio oversampling, M I 3) Arbitrary integer oversampling, Any M otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

12 Critical Sampling =M xn hn M X m n K Let M X (8) r 1 1 m hr xm r where X r 1 1 m h r x m r h x r hr m r xm r (9) (1) Frequency channels are selected for to 1 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

13 Text Based Representation The sum of the complex weighted polyphase elements TDM of M channels otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

14 Channel Bandwidths The channel bandwidths are based on the filter spectrum Spectrum Analysis Aliasing of Power into channel FDM Channel Processing Anti-alias stopband present otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

15 Integer Oversampling =M*I xn hn M X m n K Let M I K X xm r (11) r 1 I 1 m m hr I where X x m r r 1 I 1 m m h r h x I r hr m I r xm I r (12) (13) ote how the x values are stepped or shifted by an amount /I instead of prior to each computation. In addition, a post-multiplication based on is required. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB m I 15

16 Fourier and Time Delay Thin about the phase resulting from a pure time delay X 63 m n mm X m m Then for M=32, =64, I=2 M m m exp j m m X 1 2 If this time delay is applied as a circular shift in the data prior to the FFT, no post multiplication would be required! otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

17 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB The Fourier Matrix From the structure of the Fourier Matrix, here are only unique complex values but they are repeated based on he circular rotation around 2**n/ n n

18 FFT and post-multiplication Focusing on the FFT and Additional Complex Multiplication m32 Focusing on X m PP m mm or more generally X m PP m 1 Let n m M mod X 1 n (14) mm nmm m mod PP (15) nmm m mod X 1 n n m PP m nmm mod (16) otice hat this is the FFT of a circularly shifted input sequence! Therefore, to accomplish the equivalent of a post-fft complex multiplication based on m, the FFT input sequence can be reordered prior to the transform otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

19 Text Example: M=32, =64, I=2 Input data sequencing X Creating the polyphase filter elements prior to the Fourier Transform r 1 m m h r x m 32 r PP X m h r x m 32 r 64 r1 63 m m 2 PP m otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

20 Potential Application 1 FM radio stations are located every 2 Hz from from 88.1 MHz to 17.9 MHz. 99 possible radio stations RF/IF design Assume a 25.6 MHz band containing the 2 MHz bandwidth of the FM radio band. e wish to complex tune MHz radio stations consisting of 15 dummy or don t care stations, 99 useful stations, and another 14 dummy stations. In IF terms, 88.1 MHz => 15 x.2 Hz (3 MHz) or 98 MHz => 12.9 MHz. The downconverted RF is now located from 2.9 to 22.9 MHz, centered at 12.9 MHz. (mixing frequencies of 85.1 MHz for low side or 11.9 MHz for high side) Use K = 128 point transforms. Define M based on the desired sample rate for each radio channel. To be conservative, lets have a radio channel complex sampling rate of 8 Hz where each output sample is a complex I,Q pair. Then, 25.6 MHz/.8 MHz = 32 = M. (ote that we want M to be an integer.) Polyphase filter bandwidth desired: LPF passband 125 Hz stopband 375 Hz Output sample rate supports simple digital demod methods otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

21 Potential Application 2 Direct sampling of the AM radio Spectrum. Sample rate 5.12 Msps real. Bandpass filter before sampling: 5 Hz to 18 Hz. K=512 where we are interested in bins 56 (56 Hz) to 161 (161 Hz). For easy am demodulation, use a 4 Hz complex sample output rate. M = 512/4 = 128. Polyphase filter desired: LPF 1 Hz passband 3 Hz stopband at 512 Hz sample rate. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

22 Arbitrary M xn hn M X m n K 1 mm Repeating X m hr xmm r r1 (7) where X 1 mm m h r x mm r h x r1 r hr mm r xmm r (17) (18) The polyphase filter operates as expected with the input data shifted by M for successive mm computations. As before, a post-multiplication is required, but is not based on. ote: for some values of and m, no multiplication will be needed. For the remaining and m, the output is multiplied by a complex twiddle factor. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

23 Arbitrary M FFT Based on the previous results, we can form the following. m h r x mm r PP (19) r1 1 mm and X m PP m Again, let n m M mod X 1 n mm nmm m mod PP nmm m mod X 1 n n m PP m nmm mod (2) First, the polyphase taps are formed from the M bloc updated samples for each m according to Equ. (19). The resulting polyphase filter outputs (PP) are then circularly shifted based on the modulo offset of mm, and finally Equ. (2) is computed. ote that can define any desired channel spacing on to 2 and that M defines any desired integer decimation of the input sample rate otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

24 Example Problem from Text p. 234 e have a signal containing 5 FDM channels separated by 192 Hz centers containing symbols modulated at 128 Hz by square-root yquist filters with 5% excess bandwidth. Our tas is to baseband channelize all 5 channels and output data samples from each channel at 256 s/s, which is two samples per symbol. The specifications of the process are listed next and the spectrum of the FDM input signal and of one of the 5 output signals is shown in Figure 9.6. umber of Channels 5 Channels Spacing 192 Hz Channel Symbol Rate 128 Hz Shaping Filter SQRT Cosine Taper Roll Off Factor α 5% Output Sample Rate 256 Hz (2x symbol rate) otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

25 Example Spectrum Input Spectrum 5 channels spaced by 192 Hz Passband and transition bands of outer channels LPF Passband 49 * 192 Hz+ 2*64 Hz = 9,536 Hz LPF Stopband 49 * 192 Hz+ 2*64 Hz + 2*64 Hz = 9,664 Hz min. Is this the optimal sample rate for efficient signal processing? otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

26 Sample Rate and Transform e start by selecting a transform size greater than the number of FDM channels to be processed. As indicated in (9.6), the product of the transform size and the channel spacing defines the input sample rate of the data collection process. As shown in (9.7), a restatement of the yquist sampling criterion, the excess bandwidth spanned by the extra channels in the transform is allocated to the transition bandwidth of the analog anti-alias filter. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

27 Design Options Picing Magic umbers Important Factor: low-cost, efficient implementation Radix-2 or 2^n FFT preferred! Channel spacing x FFT size = 192 Hz x 64 = 12,288 Hz Filter order otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

28 Decimation Rate Sample Rate = MHz (64 x 192 Hz) Output Channel Rate = 256 Hz 12,288 Hz/ 256 Hz = 48 A decimation rate of 48 must be applied xn hn M X m n K K 64 M 48 f sin MHz f cout 256 Hz f channel 192Hz otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

29 Filter Ban Structures Maximally Decimated: 192 sps output incorrect Arbitrary M Decimation: 256 sps output correct otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

30 Polyphase Input Buffer Successive shifting and loading by blocs of 48 samples A circular queue or buffer is used. (memory addressing?) PP m h r x mm r r1 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

31 Polyphase Input Buffer Successive shifting and loading by blocs of 48 samples PP X m h r x mm r r1 1 n n m PP m nmm mod otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

32 Pre-FFT Cyclic Shifts A unique shift based on the least common denominator of K and (16). X 1 n n m PP m nmm mod otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

33 Implementation Options e have a signal containing 5 FDM channels separated by 192 Hz centers containing symbols modulated at 128 Hz by square-root yquist filters with 5% excess bandwidth. Our tas is to baseband channelize all 5 channels and output data samples from each channel at 256 s/s, which is two samples per symbol. The specifications of the process are listed next and the spectrum of the FDM input signal and of one of the 5 output signals is shown in Figure 9.6. umber of Channels 5 Channels Spacing 192 Hz Channel Symbol Rate 128 Hz Shaping Filter SQRT Cosine Taper Roll Off Factor α 5% Output Sample Rate 256 Hz (2x symbol rate) otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

34 Example Alternatives (p. 241) There are other ways to implement the previously defined problem. Section 9.2.1, p. 241, shows 5 ways. Start at Msps input End at 256 sps output Apply matched filter (MFsquare-root yquist) Use an arbitrary LPF when needed Employ channel band filters and individual rational rate resampling to achieve the final rate K=64, M=64 followed by 3-to-4 Rational Rate Change with MF K=64, M=64 with MF followed by 3-to-4 Rational Rate Change K=64, M=32 followed by 3-to-2 Rational Rate Change with MF K=64, M=48 followed by MF K=64, M=48 with MF otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

35 Option Bloc Diagrams Design 1 Design 2 Design 3 otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

36 Option Bloc Diagrams (2) Design 4 Design 5 hich option do you thin is the most efficient? The last one of course proven in the following pages. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

37 Design Option Descriptions Design 1. Channelize the FDM signal with 64-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 192 Hz, then up sample each channel by 4/3 to obtain the desired 256 Hz output sample rate, and finally pass each correctly sampled channel series through a matched filter. To reduce processing burden and cost, the matched filter is embedded in the 4/3-interpolator filter. Design 2. Channelize the FDM signal with 64-to-1 down sampling in the 64-path polyphase matched filter to obtain a per-channel output sample rate of 192 Hz, then up sample each matched filter series by 4/3 to obtain the desired 256 Hz output sample rate. Design 3. Channelize the FDM signal with a 32-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 384 Hz, then down sample by 3/2 to obtain the desired 256 Hz output sample rate, and finally pass each correctly sampled channel series through a matched filter. Here too, to reduce processing burden and cost, the matched filter is embedded in the 3/2-interpolator filter. Design 4. Channelize the FDM signal with a 48-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 256 Hz, and finally pass each correctly sampled channel series through a matched filter. Design 5. Channelize the FDM signal with a 48-to-1 down sampling in the 64-path polyphase matched filter to directly obtain a per-channel matched filter output sample rate of 256 Hz. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

38 otes On Designs The specifications of the process are listed next and the spectrum of the FDM input signal and of one of the 5 output signals is shown in Figure 9.6. umber of Channels 5 Channels Spacing 192 Hz Channel Symbol Rate 128 Hz Shaping Filter SQRT Cosine Taper Roll Off Factor α 5% Output Sample Rate 256 Hz otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

39 M=64 Polyphase Filter Pure Polyphase first stage Fsin = 12,288 Hz (64 x 192 Hz) Fpass = 64 Hz Fstop = (192-64=128) Hz possible, 96 Hz preferred Fsout = based on design (192, A, 384, 256, A) otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

40 Matched Filter Rational Rate Change The matched filter is a Square-root yquist filter. Input Sample Rate 768 Hz (varies with design 192 x 4) Channel Symbol Rate 128 Hz Shaping Filter SQRT Cosine Taper Roll Off Factor α 5% Output Sample Rate 256 Hz From yquisttestv2.m (=5 for option 1) hsqnyq=firrcos(2**m,fsymbol/2,alpha,fsample,'rolloff','sqrt')'; hsqnyqfh=nyq_fharris(fsymbol,fsample,alpha,,1)'; otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

41 Option 1 Design 1. Channelize the FDM signal with 64-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 192 Hz, then up sample each channel by 4/3 to obtain the desired 256 Hz output sample rate, and finally pass each correctly sampled channel series through a matched filter. To reduce processing burden and cost, the matched filter is embedded in the 4/3- interpolator filter. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

42 Option 2 Channelize the FDM signal with 64-to-1 down sampling in the 64-path polyphase matched filter to obtain a per-channel output sample rate of 192 Hz, then up sample each matched filter series by 4/3 to obtain the desired 256 Hz output sample rate. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

43 Option 3 Design 3. Channelize the FDM signal with a 32-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 384 Hz, then down sample by 3/2 to obtain the desired 256 Hz output sample rate, and finally pass each correctly sampled channel series through a matched filter. Here too, to reduce processing burden and cost, the matched filter is embedded in the 3/2-interpolator filter. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

44 Option 4 Design 4. Channelize the FDM signal with a 48-to-1 down sampling in the 64-path polyphase channelizing filter to obtain a per-channel output sample rate of 256 Hz, and finally pass each correctly sampled channel series through a matched filter. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

45 Option 5 Design 5. Channelize the FDM signal with a 48-to-1 down sampling in the 64-path polyphase matched filter to directly obtain a per-channel matched filter output sample rate of 256 Hz. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

46 Option Comparison This is an example of tass that DSP system engineer and designer must perform to optimize implementations (H/ or S/) and algorithm applications. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB

47 Technical Paper Review Harris, F.J.; Dic, C.; Rice, M., "Digital receivers and transmitters using polyphase filter bans for wireless communications," Microwave Theory and Techniques, IEEE Transactions on, vol.51, no.4, pp.1395,1412, Apr 23. Harris, F.; Dic, C., "Polyphase channelizer performs sample rate change required for both matched filtering and channel frequency spacing," Signals, Systems and Computers, 29 Conference Record of the Forty-Third Asilomar Conference on, vol., no., pp.1283,1287, 1-4 ov. 29. otes and figures are based on or taen from materials in the course textboo: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 24. ISB