Multirate Filtering, Resampling Filters, Polyphase Filters. or how to make efficient FIR filters

Transcription

1 Multirate Filtering, Resampling Filters, Polyphase Filters or how to make efficient FIR filters

2 THE NOBLE IDENTITY 1 Efficient Implementation of Resampling filters H(z M ) M:1 M:1 H(z) Rule 1: Filtering with M-unit delays followed by a M:1 downsampling is equivalent to M:1 downsampling followed by filtering with 1 unit delays.

3 THE NOBLE IDENTITY 2 Efficient Implementation of Resampling filters 1:M H(z M ) H(z) 1:M Rule 2: 1:M upsampling followed by filtering with M-unit delays is equivalent to filtering with 1 unit delays followed by 1:M upsampling. It is always more efficient to apply the filter at the lower sample rate

4 Downsampling Sometimes called decimation. This is poor nomenclature since decimation refers to removing 1 in 10. It is useful to think of the downsampling operation as the output of a commutator. M:1

5 Downsampling x(n) M:1 x D (n) x(n) Commutator M-1 x D (n) If x(n) has samples indexed from 0 to N-1, then x D (n) has samples indexed from 0 to (N/M)-1.

6 Upsampling Sometimes called interpolation although it is not interpolation in the conventional sense; it is just putting zeros between samples. If we wish to perform interpolation in the conventional sense, we must pass the DSP interpolated data through a low pass filter (anti-imaging or interpolation filter). In DSP, interpolation is also called 1:L zero-packing.

7 Upsampling x(n) 1:L x I (n) x(n) L-1 1:L Commutator x I (n) If x(n) has samples indexed from 0 to N-1, then x I (n) has samples indexed from 0 to LN-1.

8 Digital Receiver Options Frequency Shift Low Pass Filter Sample Sample Frequency Shift Low Pass Filter Resample Sample Low Pass Filter/Freq Shift Resample Frequency Shift Sample Resample Low Pass Filter Frequency Shift

9 Frequency Shift Low Pass Filter Sample 1 frequency shift 2 low pass filter sample 3 4 -Fs Fs

10 Receiver Structure Analog Low s(t) X 1 2 Pass Filter 3 Fs 4 e j 0 t r(n) Which can be implemented by s(t) 1 X X cos sin 0 0 t t Analog Low Pass Filter Fs Analog Low Pass Filter Fs x(n) y(n)

11 Comments Need to have two perfectly matched analog mixers two perfectly matched high order analog filters two perfectly matched analog-to-digital converters Difficult to achieve and expensive Reduce expense by performing some of these operations in DSP

12 Sample Frequency Shift Low Pass Filter Resample... low pass antialias filter sample Fs/2 Fs/2

13 frequency shift 4 low pass filter -Fs resample Fs 5 6 -Fs/M Fs/M

14 Receiver Structure s(t) 1 4 Analog Low Pass Filter 2 3 Fs Digital Low Pass Filter 5 Fs/M 6 X e j 0 n r(t)

15 Sample Low Pass Filter/Freq Shift Resample Frequency Shift low pass antialias filter sample f 0 f f s /2 f 0 f s /2

16 frequency shifted lowpass filter (bandpass filter) 4 resample f 0 5 frequency shift f0 Kfs / M 6 f s / M f s / M f s / M > two sided bandwidth

17 Receiver Structure s(t) 1 Analog Low Pass Filter 2 3 Fs Bandpass Filter 4 Fs/M X 5 6 r(t) e jm n

18 Comments No need to frequency shift signal to baseband before resampling. As long as the sampling frequency is greater than the two sided bandwidth, no information will be lost due to aliasing. This condition is sufficient even if the signal is passband although the resampling may appear to cause a frequency shift. This technique is exploited in sampling oscilloscopes which capture very high frequency signals with low sampling rates.

19 Sampling Oscilloscope Principle HF Waveform Sampled data is a time scaled version of HF waveform

20 Sample Resample Low Pass Filter Frequency Shift low pass antialias filter f 0 2 sample f 0 3 f s /2 f 0

21 Receiver Structure s(t) 1 Analog Low Pass Filter h 0(n) 2 3 Fs 5a X polyphase filter commutator 4 Note direction of indexing... h 1(n) h 2(n) h M-1(n) 5b 5c X X 5M 6 X + 7 Resample and Filter r(t)

22 resample 5a 5b 5c real real real f f f

23 Delay t=0.t/4 t=1.t/4 t=2.t/4 cancel t=3.t/4

24 Comments In the polyphase filter design we introduce deliberate aliasing by downsampling. Thus at the output of each filter, the desired signal is jumbled up with replicas of the other unwanted bands. Due to the phase delays in the separate paths through the filter, the signals in the unwanted bands will cancel at the summing node leaving only the desired signal. This structure is computationally more efficient than the previous bandpass filter design (by a factor of M).

25 Polyphase Filter Partition Let N = L*M N = Filter Length M = Resampling Rate L = Subfilter Length Note- can always zero pad to make N = L*M Place filter coefficients columnwise into an M by L matrix. Subfilters are the rows of the matrix. h h h h h h h h h h h h h h h h M + L

26 x n Polyphase Downsampler in Detail y 3:1 h 0 h 1 h 2 h 3 h 4 h n 5 0: y 0 = h 0 x 0 1: y 1 = h 1 x 0 +h 0 x 1 2: y 2 = h 2 x 0 +h 1 x 1 +h 0 x 2 3: y 3 = h 3 x 0 +h 2 x 1 +h 1 x 2 +h 0 x 3 4: y 4 = h 4 x 0 +h 3 x 1 +h 2 x 2 +h 1 x 3 +h 0 x 4 5: y 5 = h 5 x 0 +h 4 x 1 +h 3 x 2 +h 2 x 3 +h 1 x 4 +h 1 x 4 y 2 = h 2 x 0 +h 1 x 1 +h 0 x 2 y 5 = h 5 x 0 +h 4 x 1 +h 3 x 2 +h 2 x 3 +h 1 x 4 +h 1 x 4 x n h 0 h 3 h 1 h 4 + h 2 h 5 Note: outputs sample M-1, 2M-1, 3M-1.

27 Filter Design Equations Prototype Spectrum Replicate Spectrum A(dB) 0 f f s f = 200 Hz Example N A(dB ) 22 f s f f s = 20 khz A (db) = 80 db ,

28 f s N-Tap LP Filter f s/m Prototype Spectrum Replicate Spectrum at output rate Spectral Shift Replicate Spectrum at input rate 0 f s/m f BW f f K A f s N KA A( db) ( ) ; ( ) 22 f a f BW s M a f s M fractional bandwidth output sample rate f s f BW M f s a M 1 a f s M K N M 1 f M ( s A a f K ) f s K ( A ) ( A ) f N s f N s

29 N M K( A) 1 a N/M determined by quality specifications of filter N = filter size, number of ops per output point M = resample ratio, number of input points per output point N/M = Number of Ops per input point

30 Example K( A) 36. ( 80dB) a 06. N KA ( ) M 1 a 0.4 Standard filter requires 364 Ops Number of Ops/Input Independent of bandwidth Independent of sample rate Independent of filter length Dependent only on performance measures fractional bandwidth (a) and out-of-band attenuation

31 Comments This design formula lets the spectral replica overlap in the transition band so the attenuation never reaches the design value within the transition band. Need to use a filter design that has a small roll-off (say 3db per octave) in the stop band to ensure that the out of band noise does not stack up and prevent the filter from meeting specifications.

32 Upsampling T Zero Packing Spectral Period 1/T T/M Spectral Period M/T

33 Upsampling interp After low pass filtering to remove images (interpolation, reconstruction, or anti-imaging filter) T/M Spectral Period M/T Interpolation(upsampling) = zero packing and low pass filtering

34 Cooking Class Biscuit cutter

35 Biscuit Cutter Model of Zero-Packing and DSP Filtering one copy New Biscuit Cutter Biscuit Cutter four copies

36 Input spectrum replicates every interval Fs =1/T Period of spectrum with time samples every T seconds is Fs Corresponds to one pattern per drum on biscuit cutter Each rotation leaves one pattern on cooking dough biscuit cutter Input samples Fs

37 Output spectrum replicates every interval Fs =1/T Period of spectrum with zero-packed time samples every T/4 seconds is 4Fs Corresponds to four pattern per drum on biscuit cutter Each rotation leaves four patterns on cooking dough new biscuit cutter Zero Packing 4 Fs

38 To change spectral replicating interval from Fs to 4Fs, remove three spectral replicates from the biscuit cutter. Then period of spectrum is 4Fs, and zero-packed samples are replaced with bandlimited interpolated samples. modiifed biscuit cutter Low Pass Filter 4 Fs

39 DSP Approach Spectrum: input to Upsampler cirdemo Spectral Period Fs Spectrum: Input to Digital Lowpass Spectral Period MFs Spectrum: Output of Digital Lowpass Spectral Period MFs

40 Comments We have not used the Noble Identity which says that upsampling followed by filtering can be achieved by filtering followed by upsampling. Thus we should implement this upsampling with a polyphase structure to reduce the computational load by a factor of M.

41 M Polyphase Upsampler Let N = L*M N = Filter Length M = Resampling Rate L = Subfilter Length L h h h h h h h h h h h h h h h h Place filter coefficients columnwise into an M by L matrix. Subfilters are the rows of the matrix. Note - can always zero pad to make N = L*M

42 Another View x(n) M-1 D D D D D Filter Coefficients