Digital Signal Processing

Transcription

1 Digital Signal Processing Multirate Signal Processing Dr. Manar Mohaisen Office: F28 School of IT Engineering

2 Review of the Precedent ecture Introduced Properties of FIR Filters Introduced Applications of FIR Filters Introduced the Design Methods of FIR Filters Windowing Method Frequency-sampling Method east-square Method Introduced two Applications of FIR Filters Differentiators Hilbert Transforms

3 Class Objectives Introduce multirate processing Introduce integer sampling rate conversion Decimators Interpolators Introduce rational sampling rate conversion Single stage converters Multi-stage converters Introduce multirate filter realization structures Introduce sub-band processing Narrow-band filter design using rate converters Filter banks

4 Integer Sampling Rate Converters Multirate Systems Signals are sampled at different sampling frequencies E.: replacing the anti-aliasing filter with oversampling How to change the sampling rate of a signal? ( kt ) ( t ) ( kt 2 ) Drawbacks DAC and ADC introduce quantization noise and aliasing errors. So, how to convert the sampling rate in the discrete-time domain?

5 Integer Sampling Rate Converters Method : Sampling rate decimator Reduce the sampling rate by a factor of M et ( k) ( kt), k, BW fs 2 Then, the decimated signal a M ( k ) ( Mk ), k Problem: This simple method does not consider the frequency contents of the signals. To avoid aliasing: The analog signal must be limited to f s /(2M) Therefore, pass (k) through a low-pass filter ( ), fs H f f M 2M, elsewhere

6 Integer Sampling Rate Converters Method : Sampling rate decimator contd. Consider an of the FIR filters of Chapter 5, then m y( k) b ( Mk i) i i r y

7 Integer Sampling Rate Converters Method : Sampling rate decimator contd. Eample 7-: a( t) sin(2 t).5cos(4 t) M = 2, m = 2 with Hamming window. 2 Original samples (k) k Decimated samples 2 y(k) k

8 Integer Sampling Rate Converters Method 2: Sampling rate interpolator Increase the sampling frequency by a factor of The easiest way is to insert (-) zeros between the sample. ( k) Effect of zero samples Therefore, ( k / ) ( k), k,, 2,..., otherwise X ( z) X ( z) X ( f ) X ( f ), f fs The spectrum of the interpolated signal is -fold replication of (k) H ( ), fs f f 2, elsewhere

9 Integer Sampling Rate Converters Method 2: Sampling rate interpolator Consider an FIR filter from chapter 5, then m y ( k ) b ( k i ) i i k i r y

10 Integer Sampling Rate Converters Method 2: Sampling rate interpolator Eample 7-2: a( t) sin(2 t).5cos(4 t) = 3, m = 2 with Hanning window. 2 Original samples (k) k Interpolated samples 2 y(k) k

11 Rational Sampling Rate Converters Single stage converters Increase/decrease the sampling rate by a noninteger factor This can be done by using a decimator and an interpolator. The new sampling frequency is therefore f ' s M f s interpolator decimator H (z) H M (z) M y H (z) M y

12 Single stage converters contd. H (z) is a low-pass filter with cut-off frequency Rational Sampling Rate Converters y f f Therefore, min, 2 2 s s f f F M ( ) ( ) m i i Mk i y k b Mk i

13 Rational Sampling Rate Converters Single stage converters contd. Eample 7-3: a( t) cos(2 t).8sin(4 t) F /M = 3/2 with fs = 2 Hz, therefore Original samples 2 min, Hz (k) k Rate-converted samples 2 y(k) k

14 Multistage Converters Multistage converters When or M is large It is hard to implement the conversion using a single convertor. In this case, M r i M i i A multistage sampling rate convertor with r = 2 stages. y

15 Multirate Filter Realization Structures An interpolator with >> Most of the samples processed by the low-pass filter are samples Many of the operations are multiplications by Note that we insert (-) zeros between the samples of the input signal A decimator with M >> Only every M-th sample of the filter s output is used However, every sample of the input is processed. As such, Efficient implementations of the interpolator/decimator are required.

16 Multirate Filter Realization Structures Polyphase interpolator The input-output relation is given by: m y ( k ) h ( r ) ( k r ) r k r Condition : Suppose that (m+) is an integer multiple of. Otherwise, pad h(k) with zeros so that condition is satisfied. That is, p m, for some integer p As a result, The interpolator computations are reconstructed by using subfilters depending on the value of k.

17 Multirate Filter Realization Structures k q ( ) ( ) ( ) m r k r y k h r k r ( ) ( ) ( ) p i p i q i y q h i h i q i

18 Multirate Filter Realization Structures k q m y ( k ) h ( r ) ( k r ) r k r p y( q ) h( i ) i p h( i ) q i i q ( i )

19 Multirate Filter Realization Structures k q m y ( k ) h ( r ) ( k r ) r k r p y( q ) h( i ) i p h( i ) q i i q ( i )

20 Multirate Filter Realization Structures Polyphase interpolator contd. Eample 7-5: m = 8, = 3 p = 3. ( k) z z h() h(3) y ( k) h(6) h() h(4) h(7) y ( k) 3 f s y( k) h(2) h(5) y ( k) 2 h(8)

21 Multirate Filter Realization Structures Polyphase interpolator contd. We have ( -) cases, Case : Case : p y( q) h( i) q i i p y( q ) h( i ) q i i g ( k) h( k), k p g ( k) h( k ), k p Therefore, The impulse response for the i-th polyphase filter is given by: gi ( k ) h ( k i ), i, k p

22 Multirate Filter Realization Structures Polyphase interpolator contd. Back to eample 7-5: m = 8, = 3 p = 3. y ( k) ( k) y ( k) 3 f s y ( k ) y ( k) 2 For each output sample, Instead of (m + ) multiplications, only p multiplications are required.

23 Multirate Filter Realization Structures Polyphase decimator Similar to the interpolator, decimator can be reconstructed using polyphase filters. y ( k) ( k) y ( k) y( k) ym ( ) k gi ( k ) h ( Mk i ), i M, k p

24 Subband Processing Narrowband filters A sharp filter whose passband/stopband is much smaller than f s Require very high-order FIR filters. This implies high storage, computations requirements. Also, high-order filters are sensitive to finite word length effects. Multirate narrowband filter using rate conversion Goal: Design a P filter with a cutoff frequency of F << f s Step : Decimator increases the relative cutoff freq. of the filter. Decimator decreases the sampling rate by a factor of M. Now, F becomes MF. Select a suitable value of M. Step 2: Design a filter with cutoff frequency of MF. Easy to imple ment. Step 3: Interpolator restores the original sampling frequency. Cutoff frequency of the designed filter becomes F (required value)

25 Subband Processing Multirate narrowband filter using rate conversion contd. How to select M? Set M f 4F s H ( f ), f F, F f fs / 2 y y G( f ), f F M, F M f fs / 2

26 Subband Processing Multirate narrowband filter using rate conversion contd. Eample 7.6: F = f s /32 M f 8 4 s F G(z) and H M (z) are FIR filters of order m = 8. H(z) is an FIR filter of order m = 24. Both designs have same storage requirements. However, multirate design has superior performance in terms of the width of the transition band..2.8 Magnitude responses Ideal Fied-rate Multirate A(f) f/f s

27 Subband Processing Filter banks It some applications, Frequency bands are processed differently A band might be more important than another/others Therefore, a signal is synthesized into several bands before being processed. Called frequency division multipleing. This can be done using filter banks. (equivalent to an allpass filter).5 A filter bank A A A 2 A 3 A(f) f/f s

28 Subband Processing Filter bands contd. H (z) G (z) ( k) H subband (z) G (z) + processor y( k) H N- (z) analysis filter bank G N- (z) synthesis filter bank Analysis filter bank Decomposes the overall spectrum into N subbands. Synthesis filter bank: Recombines the subsignals into one output signal y(k) A decimated and interpolated implementation of FB is possible.

29 Subband Processing Filter bands contd. Uniform DFT synthesis and analysis N Herein, the assumption of real-valued input signal can be relaed. ( k) ( k ) ( k ) M M M H N W N k W N H N + H N k ( N ) W N ( i) ( i) W N i W N ( k) ( k) i ( N ) W N ( k ) N synthesis using DFT filter bank analysis using DFT filter bank Remember: The shift property. DTFT ep( jk 2 FT ) ( k ) X ( f F ) i i

30 Subband Processing Filter bands contd. Eample 7-7: Signal synthesis N = 4, f s = Hz, p = 64 points, signals are band-limited to f f s /4. Signals are defined as follows for f i = if s /p for i < p with zero phase. X ( i) cos 2 F f i X ( i) f / F X i 2 ( i) sin f F i 2 ( ) / 3 i X i f F A(f) Subsignal magnitude spectra f (Hz) 3 2

31 Subband Processing Filter bands contd. Eample 7-7: Signal synthesis Step : obtain the time domain signals Step 2: upsample signals by a factor of N = 4. Step 3: ( k) ( k) 2 ( k) 3 ( k) M M M M H(z) H(z) H(z) H(z) Filter signals by an anti-aliasing filter of order m = 2 (or other.) Step 4: Synthesis the signal Step 5: ( k) 2 3 ( k) W4 ( k) W4 2( k) W4 3( k) ( k) j[ ( k) ( k)] 2 3 Find the spectrum of the synthesized signal (k) W 4 2 W 4 3 W 4 + ( i)

32 Subband Processing Filter bands contd. Eample 7-7: Signal synthesis Signal duration becomes (p N) = 64 4 = 256 samples ( k) ( k) j[ ( k) ( k)] 2 3 Composite signal )} Real{(k).5 Imag{(k)} k k

33 Subband Processing Filter bands contd. Eample 7-7: Signal synthesis Composite magnitude spectrum A A A 2 A 3 A A(f) f (Hz) Signals have been shifted and centered at F i = if s /N for i < 4

34 Subband Processing F c f 2N s 4 Composite magnitude spectrum A A A 2 A 3 A A(f) f (Hz)

35 ecture Summary Introduced multirate processing Introduced integer sampling rate conversion Decimators Interpolators Introduced rational sampling rate conversion Single stage converters Multi-stage converters Introduced multirate filter realization structures Introduced sub-band processing Narrow-band filter design using rate converters Filter banks