Module 9: Multirate Digital Signal Processing Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering &

Transcription

1 odule 9: ultirate Digital Signal Processing Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications The University of New South Wales Australia

2 ultirate Digital Signal Processing The increasing need in modern digital systems to process data at more than one sampling rate has led the development of a new sub-area in Digital Signal Processing known as multirate processing The two primary operations in multirate processing are: Decimation Down sampling Interpolation Up sampling ELEC97

3 The processes of decimation and interpolation are the fundamental operations in multirate signal processing Decimation: A sampling rate decreaser is shown below. We shall confine our attention to a decrease by an integer factor e.g. 3 xn ym n xn f s ym f s / m 3; f s sampling frequency The output signal ym is obtained by taking every th sample of the input signal. If 3, we should just take every third sample of xn to form the desired signal ym. 3

4 Decimation xn f s 3 ym f s / Obviously, it only makes sense to reduce the sampling rate if the information content of the signal we wish to preserve is band limited to fs/6. 4 ELEC97

5 The diagram below shows a block representations of a times decimator. The signal xn is first passed through a lowpass filter that attenuates the band from {fs/}/ to fs/ to prevent aliasing. xn f s Digital Lowpass filter wn f s ym f s / db fs/ fs/ 5 f

6 Image X Input spectrum fk Digital low pass filter cut-off frequency / f s fk f s / W Filtered spectrum W fk fk x Spectral interpretation of decimation of a signal: from 6k to k 6

7 Such a processing requirement may arise, for example, when a speech signal is oversampled at fs3 k. Since we are interested only in a band of -4 k, the sampling rate can be reduced to 8 k, so the first step in the decimation process has to be the digital filtering of the signal xn so as to ensure that xn is band limited to fs/8. xn f s 3 k Digital Lowpass filter wn f s 4 ym f s /8 k 7

8 Should we use IIR or FIR for the lowpass filtering required? Using an IIR filter in this case has an obvious shortcoming. We cannot take advantage of the fact that we only have to compute every Nth output, since previous outputs are required to compute the th output. Thus no saving is realised. On the other hand, using an FIR filter, in this case implies that we can do our computations at the rate of fs/. Thus, using an FIR filter in the decimation process will lead to a significantly lower computation rate. Another advantage of using an FIR filter is the fact that we can easily design linear phase filters and this is desirable in many applications. 8

9 Interpolation The process of interpolation involves a sampling rate increase xn f s The sequence xn was derived by sampling xt at a sampling rate f s and we want to obtain a sequence yn that approximates as closely as possible the sequence that would have been obtained had we sampled xt at the rate Lf s. Interpolation involves inserting between any samples xn and xn- an additional L- samples. L ym Lf s 9

10 Interpolation Example xn f s L LPF ym Lf s X fs/ fs/ fs fs 3fs 4fs f Y L4 /4 /4 9/4 Sampling frequency of ym 4f s ; Signals must be band limited to f s

11 We observe that to go from X to Y, we have to pass xn through a lowpass digital filter designed at the Lfs sampling rate that attenuates sufficiently any frequency components above fs/.

12 Interpolator xn f s L wm Lf s Digital Lowpass filter ym Lf s Insert L- eros Example: xn{,.9,-.5}; Let L 3: wm {,,,.9,,,-.5,,} The lowpass filter joins all the xn samples of wm to produce a waveform as if xn had been n sampled at Lfs wm ym m m

13 Interpolator We assume that behind each xn there are L- ero samples when we computing an output wn Note that for each sample of xn {see previous slide}, three output samples yn are obtained Obviously the same reasoning that led us to believe that FIR filters are preferable in the decimation process holds here also. 3 ELEC97

14 Example xn f s 8k L3 wm Digital Lowpass filter ym f s What should be the cut-off frequency of the digital lowpass filter? What should be the value of f s? 4 ELEC97

15 Sampling Rate Conversion by Non-Integer Factors In some applications, the need often arises to change the sampling rate by a non-integer factor An example is in digital audio applications where it may be necessary to transfer from one storage system to another, where both systems employ different rates 5 ELEC97

16 Example An example is transferring data from the compact disk system at a rate of 44. k to a digital audio tape DAT at 48 k This can be achieved by increasing the data rate of the CD by a factor of 48/44., a non-integer In practice, such a non-integer factor is represented by a rational number, that is a ratio of two integers say L and The sampling frequency change is then achieved by first interpolating the data by L and then decimating by 6

17 Note: It is necessary that the interpolation process preceeds decimation,otherwise the decimation process would remove some of the desired frequency components CD DAT 44. k 48 k To change the sampling rate we require: L Therefore if we upsample by L6 and then down sample by 47, we achieve the desired sampling rate conversion. 7

18 Figure below shows that the sampling frequency change is achieved by first interpolating the data by L and then decimating by. xn f s L wm Lowpass Filter hm Lf s pm Lf s Lowpass Filter hm qm Lf s yk Lf s / Interpolator Decimator The two Digital Lowpass Filters, hm and hm can be combined into a single filter since they are in cascade and have a common sampling frequency. If > L the resulting operation is a decimation process by a non-integer < L the resulting operation is an interpolation 8

19 Summary: Sampling Rate Conversion by Non-Integer Factors xn f s L wm Lf s Lowpass Filter hk qm Lf s yk Lf s / The lowpass filter that we require is the one that has a cut-off frequency: c f c min, L min fs Lf s, 9 ELEC97

20 Example Figure below shows sampling rate conversion by non-integer factors. Calculate the values of L and xn f s 8k L wm Lowpass Filter hk qm yk f s k ELEC97

21 Decimation by Aliasing term ] Y [ X X j j Y [ X e X e ] j j [ X e X e ] [ X X ] Aliasing term Stretch X by a factor to obtain X/ xn Case Extreme Aliasing / X X/ ynxn X{/} The spectrum is stretched by down sampling

22 Decimation by xn ynxn Case X No Aliasing / / X/ Y [ X X ] X{/}

23 Interpolation by xn yn X y n x[ n / ] n, ±, ± 4,... otherwise Y Y [ X ] Y [ X e [ X ] j ] / The spectrum is The upsampled spectrum compresseed by has compressed images upsampling of X 3

24 Example An input signal xn with spectrum X is shown below. The input signal is applied to the system shown below. Sketch X, W, V, Y against. xn 3 wm 3 vn yn X 3 / 3 4 / 3 ELEC97

25 X /3 /3 W 3 /3 V /3 3 /3 3 Solution Note: By using a lowpass filter, we can eliminate the images and extract the original spectrum X. V is a compressed version of W Y /3 5 /3

26 Exercise A signal xn has a spectrum X as shown below. The signal is applied to the system shown below. The ideal lowpass filter has a gain factor of in the passband and a cut-off frequency /3.Sketch X, W, V, Y against. X xn 3 wm vn yn /3 /3 6 ELEC97

27 odulation and Sampling rate conversion Example: xt is the input signal for the system shown below. The analogue signal xt has the spectrum Xf given by:.9k f.k X f.k f.9k elsewhere xt xn pn A/D X Sampling 5 Period cosn T.4 ms ω 3 vn yn is an ideal lowpass filter gain with cut-off frequency f c 5. Sketch, one above another, X, P, V, Y against. Note : 7 P [ X X ]

28 f s.5k; ; 5/.5 3 c. T Solution Xf X /T f k /T P /X /X /T V Lowpass filter -.. /T/ /T Y 8

29 Example: below. xn xn is the input signal for the system shown If xn.5 δnδn.5δn-, show that / / 4 / 4 S / / 4 / 4 elsewhere bn an dn X G 7 en rn sn X cosn ; cosn G 4 < < 7

30 Solution { } elsewhere G B D A A A A B j e X X n n n n x 4 / 4 /. 4 cos 4 cos cos A - cos -j e - δ δ δ cosn X xn bn G an 7 dn en X sn rn cosn 4 3 ; 4 3

31 Solution 8 / 8 / 8 / 7 / 4 / 4 / 4 / 7 / 7 / 7 } {. 7. R elsewhere D D E R cosn X xn bn G an 7 dn en X sn rn cosn < 7 7 < 4 4 G 3

32 Solution { } elsewhere S R R R R S 8 / / 8 / / cosn X xn bn G an 7 dn en X sn rn cosn elsewhere S /4 /4 /4 /4 S /4 3/4 /4-3/4 / 4 3 3

33 Decimation by xn ynxn It can be shown that Y is given by: Y X k e j k Consider ; Substitute in the above equation Y [ X X ] 33 ELEC97

34 xn Interpolation by L L y n x[ n / L] n, ± L, ± L,... otherwise It can be shown that Y is given by If L then Y X L Y X 34

35 ultistage design of decimator and interpolator. By designing decimator/interpolator in multistage as cascade the complexity of the overall system can be reduced significantly. For example, a decimator of factor L can be designed as below where LL *L It is efficient because the filter required in the decimation of L is much longer than the filters required in the decimation by L and L. xn L L 35 ELEC97

36 Example: Converting a signal with Fsk to Fs4. For direct implementation down sampling by 3, because the filter requires sharp cut-off frequency it requires 87 th order filter. Number of multiplications 88*/3738 In multistage implementation down sampling first by 5 and then by because the cut-off frequencies are not that sharp it requires 93 th and 3 th order filters. Number of multiplications 94*/5 3*8/76 36

37 Polyphase decomposition As discussed earlier in multirate processing if FIR filter is used then lots of computations can be avoided. Consider a sequence, x[n] with a -transform X: If a subsequence is defined as below x k [ n] x[ n k], k These sub-sequences are called polyphase components and n n X x[ n] n X k n n x k [ n] n n n 37 x[ n k] n

38 Example x[n]{,,3,4,5,6,5,4,3,,} If it is decomposed into two polyphase components, x [n]{,3,5,5,3,} x [n]{,4,6,4,} xn - x [n]x[nk]x[n]{,3,5,5,3,} x [n]x[nk]x[n]{,4,6,4,} 38

39 Example Lets consider the 5 th order filter, If it is decomposed into two polyphase components, },,,,, { ] [ h h h h h h h h h h h h n h E E h h h E h h h E

40 Note the following identity L x[n] L w[n] y[n] L x[n] w'[n] y[n] L L L X Y X W ' L L X Y X W

41 Note the following identity x[n] w[n] y[n] x[n] Z w'[n] y[n] k k j k k j e X Y e X W similarly ' k k j k k j k j e X Y e X e Y X W

42 x[n] y[n] Now consider a decimation by a factor of for a sequence of x[n], assume a 5 th order filter. Computationally efficient polyphase implementation of decimation },,,,, { ] [ E E h h h E h h h E h h h h h h n h Using the following decomposition xn E - E yn

43 That is equivalent to xn E yn - E This can be implemented as below using the identity xn yn E - E This is very efficient as the filtering is performed in the down sampled signal 43

44 In the general case the computationally efficient structure for decimation is as follows xn - E yn E E E - where E, E,... E - are the polyphase decomposition of the filter. 44

45 Similarly, the general case for the computationally efficient structure for interpolation is as follows xn E L yn - E L - E. L.. E - L where E, E,... E - are the polyphase decomposition of the filter. 45

46 Application of multirate processing Design of phase shifters with rational fraction Analysis-synthesis systems with two- Channel Quadrature-irror Filter Bank ultilevel Filter Bank Regular Binary Subband tree Dyadic Subband Tree 46 ELEC97

47 Design of phase shifters with rational fraction Design a delay filter with a delay of d, dk/lt x where T x is the sampling time of the signal xn. xn L Low pass filter k sample delay Exercise: A signal is sampled at. Draw a block diagram of a system to delay the signal by.6 milliseconds.6 samples. L 47 ELEC97

48 Perfect reconstruction analysis-synthesis systems Analysis synthesis systems are used in several applications. Analysis filters split the signals and any processing can be done. Then synthesis filters reconstruct the signal. Several applications require perfect reconstruction analysis synthesis filter band except for a delay. 48 ELEC97

49 Example: Express the output yn of Figure below as function of the input xn. By simplifying the expression derived show that yn xn- xn vn qn Figure - - Solution p n P wn vn x n P - X pn W xn V pn [ [ P X - P- X X- ] wn - X- - P- ] W W rn [ P un yn - [X P- ] X ]

50 xn - yn - X Y W R Similarly - Q U ] [ ] [ ] [ ; / X X X X R U Y X X Q U X X W R X X V Q V Q n v n q xn - - vn wn rn qn un yn pn The structure shown is a perfect reconstruction system. The original signal xn is reconstructed. 5

51 Figure below shows the -channel generalisation of the structure shown in the previous slide xn channel yn ere yn xn- 5 Number of channels

52 Two-Channel Quadrature-irror Filter Bank In many applications, a discrete-time signal xn is first split into a number of subband signals by means of analysis filterbank The subband signals are then processed and finally combined by a synthesis filterbank resulting in an output signl yn 5 ELEC97

53 x n v n y n F s n yn x n v n y n F D e c o m p o s i t i o n Channel R e c o n s t r u c t i o n If the subband signals are bandlimited to frequency ranges smaller than that of the original input signal, they can be down sampled before processing After processing, these signals are up-sampled before being combined by the synthesis filter bank into a higher rate signal The combined structure employed is called a Quardrature- irror Filter QFbank. 53

54 x n v n y n F s n yn x n v n y n F D e c o m p o s i t i o n Channel R e c o n s t r u c t i o n If the down-sampling and the up sampling factors are equal to the number of bands of the filter bank, then the output yn can be made to retain some or all of the characteristics of the input xn by properly choosing the filters in the structure. In this case, the filter bank is said to be critically sampled filter bank 54

55 irror Image Filters Let h n be some FIR lowpass filter with real coefficients. The mirror filter is defined as h n- n h n Therefore - j e j e j e j e This demostrates the mirror image property of and about /. ence justifying the name quadrature mirror filters QF 55

56 Amp..5 QF Filters Lowpass ighpass Re.5 4 Frequency 56 The pole-ero patterns are also reflected about the imaginary axis of the - plane

57 Analysis of the two-channel QF Bank x n v n y n F p n s n ^ sn x n v n y n F p n D e c o m p o s i t i o n R e c o n s t r u c t i o n In subband coding applications, v n and v n are quantised, encoded and transmitted to the receiver. We assume ideal operation here, with no coding and transmission errors, so we focus on the analysis and synthesis filters. 57 ELEC97

58 ] [ ] [ ˆ ˆ...7 ] [ Similarly...6 ] [ : Substituting in ] [. : and Y : ]...3 [ V : S F F S F F S P P S F S S P F S S P F X X F Y P From V sampler Up X X sampler Down F Y P S X Aliasing term This is precisely due to aliasing caused by sampling rate alteration 58

59 9 and can be achieved by the selection : for all Perfect reconstruction requires ] [ ] [ T where ˆ F F This F F F F F F F F S S T S The remaining function T in equation 8 denotes the quality of the reconstruction. If this is a merely a delay I.e. T -k, the filter bank performs perfect reconstruction. is due to aliasing 59

60 Sˆ T S with T [ ] If T is an allpass function T d indicating that the then S ˆ output of d S the QF bank has of the input assume d and the filter bank is the same magnitude as that said to be magnitude preserving. If T is linear phase, then the filter bank is said to be phase preserving If an alias free QF bank has no amplitude and phase distortion, then it is called a perfect reconstruction PR QF. In such case T d -k, resulting ^ S d -k S Which in the time domain is equivalent to: sn d sn-k For all possible inputs, indicating that the reconstructed output sn is a scaled, delayed replica of the input. 6 ^ ^

61 From 9 F and F - - {Alias free condition} For QF bank - Combining the above three conditions, we get F and F Equation indicates that if is a lowpass filter then F is also a lowpass filter and F is a high pass filter. Using equations and we obtain T / [ ] / [ -] d -k 6

62 Example Consider a two-channel QF bank with the analysis filter given by - The second analysis filter is therefore -- - and the corresponding synthesis filters for an alias-free realisation F - - F - --[- - ] - 6 ELEC97

63 ultilevel Filter Bank It is possible to develop a multiband analysis/synthesis filterbank by iterating a two-channel QF bank. oreover, if two band QF bank is a perfect reconstruction type, the generated multiband structure also exhibits perfect reconstruction property 63 ELEC97

64 Example - 4 k cd 3 Level Decomposition sn -4 - k cd a ca.5 -. k cd3 Level ca Level - 5 ca3 ca3 F Level 3 ca F b cd3 F ca F cd F sn 3 Level Reconstruction 64 cd F

65 Regular Binary Subband tree The two-channel filterbank divides the input spectrum into two equal subbands, yielding the low L and the high subbands. This two-band QF split can again be applied to the L and half-bands to generate the quarter bands. When this procedure is repeated K times, K equal bandwidth subbands are obtained. This approach provided the maximum possible frequency resolution of / K within K levels. This spectral analysis structure is called a K- level regular binary tree. 65

66 Regular Binary Subband tree. For K 3 the regular binary tree structure that employs QF and the corresponding frequency band split are shown in Figure 3a and 3b. Figure 3c shows the frequency diagram for the 8-band regular binary subband decomposition. We assume that the filter outputs are computed for each input block of 8 samples, which will result in one new subband sample generated for each maximally decimated subband and for each input block. 66

67 A regular tree structure for K 3; L a L L L L L L LLL LL LL L LL L L agnitude Frequency Time- Frequency Diagram b / C / Frequency Time x sampling period

68 LLL L L a LL L Dyadic Subband Tree L agnitude /4 / Frequency b / Frequency f3 t3 8 6 Time x sampling period In many applications all of the subbands of the regular tree may not be required and as a result some of the fine frequency resolution subbands can be combined to yield larger bandwidth frequency bands. This implies the irregular termination of the tree branches. One of the irregular tree structures is called dyadic, or octave band tree. It splits only the lower half of the spectrum into two equal bands at any level of the tree. Therefore, the higher half-band component of the signal at any level of the tree is not decomposed any further. 68 t c t f f

69 Example In p u t BW 6 k Sampling Rate i gh ban d Lo w ban d -4 QF p ai r Decomposition Tree Structure for Audio Coding

70 Summary Decimation and Interpolation Sampling Rate Conversion by Non-Integer Factors ultistage design of decimator and interpolator. Polyphase decomposition and efficient decimation and interpolation Perfect reconstruction analysis-synthesis systems Design of phase shifters with rational fraction Two-Channel Quadrature-irror Filter Bank ultilevel Filter Bank Regular Binary Subband tree Dyadic Subband Tree 7 ELEC97