ECE 6560 Multirate Signal Processing Chapter 11

Transcription

1 ultirate Signal Processing Chapter Dr. Bradley J. Bauin Western ichigan University College of Engineering and Applied Sciences Department of Electrical and Computer Engineering 903 W. ichigan Ave. Kalamaoo I,

2 Chapter : Cascade Integrator Comb Filters. A ultiply Free Filter 36. Binary Integers and Overflow ultistage CIC Hogenauer Filter Accumulator Bit Width Pruning Accumulator Width CIC Interpolator Example Coherent and Incoherent Gain in CIC Integrators 359 Hogenauer, E.;, "An economical class of digital filters for decimation and interpolation," Acoustics, Speech and Signal Processing, IEEE Transactions on, vol.9, no., pp. 55-6, Apr 98.

3 A ultiplier Free Filter A filter with a rectangle-shaped impulse response is also called a boxcar or a sliding average filter. It performs a filtering task without multiplies. The quality of the filtering is not very good since the spectral response of the boxcar filter exhibits only 3-dB attenuation. The simplicity of one form of implementation is so attractive we are drawn to this filter even though the filtering performance is not very good. We will simply have to find a way to improve its performance. 3

4 4 Frequency Response and Implementation (infinite sum) 0 exp n n w i e H n n n w i n w i e H exp exp 0 0 exp exp n n w i w i e H 0 ' 0 ' exp exp n n n w i n w i e H ' exp exp exp 0 ' 0 n w i w i n w i e H n n

5 5 Frequency Response and Implementation (infinite sum) sin sin exp exp w i w i w i w i e H sinc sinc exp sin sin exp w w w i w w w i e H w i w i e H exp exp w Nulls at: 0 exp exp n n w i w i e H

6 Frequency Response and Implementation (textbook) H sin sin y k 0 n xn k k N First Null at: Note: The frequency response approximates a circular sinc function 6

7 Cascaded Filters As the sidelobes are still a problem, what happens if we cascade multiple Boxcar Filters? The sidelobes get small and the main lobe gets more sloped. H e 4 sinc w sinc w 4 Solved one problem 7

8 8 Boxcar Equivalent: Comb-Integrator (Finding another way with math equivalents) 0 k k n x n y x k n x n x k n x n y k k 0 0 n y x n x x n y n x n y 0 k k n x n y Let and An iterative update equation can be defined as The Boxcar implementation as a comb and integrator filters

9 Comb-Integrator Response Y e X e Y e X e e e Y y n xn x yn Y X X Y X Y X e e e j w e j w e e j sin w j sin w j w j w e e j w j w j e w j w e j w e sinw sinw Equivalent Spectral Domain 9

10 ATLAB Response Function sin/sin or sinc/sinc If the signal bandwidth is very narrow, it works. What would happen if you decimated by? The sinc/sin eros alias/align to the 0 frequency. Band edge power = db st Null Passband Power (db) Power (db) Frequency Frequency 0

11 LOOKING AT THE COB AND INTEGRATOR

12 Comb Only Response comb n xn x Comb X Comb Comb e X e Z X X CombZ X Combe e X e e j j j e w e w e w e w j e j sin w periodic eros comb teeth? Null specific frequencies

13 Comb Filter A marginal filter for signal Narrow shaped passbands sin w, for w k k k w Nulls at k = integer 0 sin w, for w k w k Peak and null locations 3

14 Integrator Only Response e Cint e X Cint cint Cint Cint X Cint e X n n0 n0 X x n e e e j w j w e e X n e j w j e e w sinw j w j e w sin For 0 <= w <= at w = 0 and w = Sinc function adjustment agnitude at w = ± 4

15 Combining the Comb and Integrator Combining Spectral Responses Infinity and ero combine to be a constant, otherwise comb filter elements persist with periodicity. Comb e X e w j e j sin w Cinte X j e e w sinw j Y e X e e e j sin w j sin w j e w j w e j w e sinw sinw 5

16 CIC Filter Structure Only two summation (adds) are required. Original Boxcar required - summations An element memory is still required (but we haven t decimated yet) 6

17 ATLAB Response Function sin/sin or sinc/sinc prior to decimation by Power (db) Frequency 7

18 8 CIC Alternate Derivation Recognition of mathematical sequences X X X X Z Y 0 k k n x n y 0 k k H 0 H H

19 CIC Concern The filter response is a composite response, but we prefer to treat it as a cascade of an FIR (numerator) and an IIR (denominator) filter athematically this has problems due to the implied canceling of the ero and pole at w=0! (See the comb plot) H 9

20 Architecture Impulse Responses (Identical output, different internal results) 0

21 Impulse Response Comb-Integrator Filter operates as expected Integrator-Comb Filter operates, but continues to recalculate input even after the filter has stabilied at a steady-state value. Internal node values do not return to ero with a ero input

22 Architecture Step Responses (Identical output, different internal results)

23 Step Response Comb-Integrator Filter operates as expected Integrator-Comb Integrator must continue summing toward infinity! The comb filter output will stabilie to a steady state output. A stable input does create a stable output, but an internal state goes to infinity! Can this approach be acceptable? Hint: if the number system is circular the distances are always the same! It works for a circular integer field! 3

24 4 We have identified a concern about register overflow in a CIC filter. To help understand the overflow we now review a number of ways binary numbers are used to represent integers. Unsigned Integers Weights and bit values. Signed Integers (Two s Complement) Binary Integers and Overflow 0, ; 0 i b i i a i a N 0 0 a a a a N b b b b 0, ; 0 i b i i i b b a a a N 0 0 a a a a N b b b b

25 Two s Complement Overflow Figure.7 presents the overflow behavior of a s-complement binary counter. The overflow is, as expected, periodic. The unique behavior of the overflow is that the difference between points in the counter (or on circle) is correct even if the counter has experienced an overflow. It is well known that intermediate overflows of a s-complement accumulator leads to the correct answer as long as the accumulator is wide enough to hold the correct answer. 5

26 CIC Filter Example CIC with =4 & x(n)= The integrator-comb architecture can function! Note the structure if decimation were to occur. 6

27 ultistage (Cascaded) IC If one works, why not cascade a number of IC stages? 7

28 8 For one stage For two stages For K stages ultistage (Cascaded) IC Response H H K H sin sin w w e H sin sin w w e H K w w e H sin sin

29 Impulse Responses =0 Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, ultirate Signal Processing for Communication Systems, Prentice Hall PTR, 004. ISBN

30 Frequency Response Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, ultirate Signal Processing for Communication Systems, Prentice Hall PTR, 004. ISBN

31 Frequency Response Comments The rect to sinc response only provided 3 db attenuation of the first sidelobe Each successive stage is provides an additional 3 db. Attenuation in db is multiplied by the number of stages. The passband is shaped by a smooth curve The 3dB point is getting smaller as K increases The response roll-off will require post-cic compensation The stopband nulls are getting broader, providing wider notches in the spectrum! 3

32 Frequency Response Detail 3

33 CIC Suppression of Example 33

34 CIC Null Suppression atlab Chap_.m and Chap_3.m ultistage CIC Band edge power = db 0 0 st Null Passband Power (db) -60 Power (db) Frequency Frequency 34

35 CIC Suppression Zones This capability is similar to having don t care filter ones in Chapter 7. 35

36 CIC Suppression Zones For a narrow bandwidth replicated signals, the suppression ones provide useful filtering! 36

37 Suppression Zones For what operations do with discover replicated one activities ost obvious interpolation the ones make good filters! Useful for interpolate by. What about aliasing ones when decimation is applied. The ones would ero any signals that would alias to the narrow bandwidth at the origin. again a useful filter! Useful for decimate by. 37

38 Interpolation and Decimation The CIC characteristics have been studied to this point. If they can be used to replace interpolation filters from Chap. 7 interpolation-filter structures, what will the interpolator do? The same goes for filter-decimation structures, what will the decimator do? 38

39 Hogenauer Filter Using a CIC filter as the filter for interpolation-filter or filter-decimation Interpolation-Filter Filter-Decimation 39

40 Hogenauer Filter By reordering the CIC components and using a rate change factor identical to the comb delay, the noble identity can simplify the architecture. For a rate change of, the nulls in the spectrum will eliminate signals (interpolated or aliased) about a specified narrow bandwidth at the location of the nulls. 40

41 ultistage Filter Decimation Example Basic Cascade Reorder Noble Identity 4

42 One ajor Concern Bit Widths The comb stages must have sufficient input bit sies to insure that the correct output exists (while allowing overflow). The example had three successive integrators! What is the potential filter gain? We must have enough bits for the input data and the filter gain b Gain Gain H K H bfilter bdata ceil log b K ceil K log filter data K 4

43 Example: K=, =0, bdata=7 b filter GainH 0 K 400 log 0 7 ceil ceil Time series output using a maximum cosine value 43

44 If too few bits are allowed Non-sinusoidal output! 44

45 Pruning the Accumulator Width The bit-width computation provide the full bit-width for every stage. This is overkill many stages do not require the full bit precision. Specific waveforms cause maximums in each of the processing stages. If these are know, the maximum bit representation for the stage can be used instead of the filter maximum. 45

46 Up-Sampling CIC Figure.7 is a block diagram of a 4-stage CIC up sampling filter. The output of each integrator in the chain is identified and is available at the indicated tap points. Yk X 4k k 46

47 CIC-Interpolator: Integrator Inputs Hogen6b example: 6-tap, 3 stage 47

48 aximum Signal Gain Based on the impulse responses x n signy k k n 48

49 Resulting aximum Integrator Responses Note that each integrator is maximied based on a different input signal. To define the bit widths needed, use the maximum for the stage! Comb filters + bit per stage (adder) Integrators bits based on max gain (8,, 4, 8) 49

50 aximum Integrator Output 50

51 aximum Response per Stage 5

52 Bit Growth 8 5

53 Down-Sampling CIC Figure.4 is a block diagram of a 4-stage CIC down-sampling filter. Pruning of this process will by performed by discarding lower order bits in each accumulator. The discarded bits will be treated as additive noise to each integrator. Our interest here is the noise gain from each integrator and from each comb filter to the output port. The input of each integrator in the chain is identified and is available at the indicated input points from which we will determine the noise gains. Y X k k 4 k 53

54 CIC-Decimator Integrator Outputs 54

55 Noise-Power Gain N n0 g n Out ( k ) k k k N G k Example Noise Gain (referred to output) 55

56 Pruning the LSB The noise gain from the later stage would be increased if fewer least-significant-bits are maintained. But since there contribution is less, the level may be increased without significant increase in the system noise growth, 56

57 Bit Decrease 57

58 CIC Interpolator Example Polyphase interpolate-filter by a factor of 5 Shape the baseband input CIC up convert by a factor of 6 Hogen6b.m and hogan6c.m Resulting interpolation rate is 5 x 6 = 80! 58

59 Shaping Filter Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, ultirate Signal Processing for Communication Systems, Prentice Hall PTR, 004. ISBN

60 Shaping Pulse Responses Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, ultirate Signal Processing for Communication Systems, Prentice Hall PTR, 004. ISBN

61 Composite Output Spectrum Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, ultirate Signal Processing for Communication Systems, Prentice Hall PTR, 004. ISBN

62 ATLAB Simulation See Chap_8.m: Cascaded Nyquist shape interpolator (x5) and 3 stage CIC interpolator (x6). Total interpolation (x80) fsymbol = ; fsample = 5; fsample = fsample * 6; 3-stage CIC interpolator Input: Impulse response or ASK symbols 6

63 Author s Examples Hogen6a Hogen6b Hogen6c Hogen50 /Chap_5 Add_ 63

64 Additional Simulation Chap_6.m odified hogen6a Chap_7.m hogen6a decimator Chap_8.m Section.5 example interpolator Notes on Nyquist filter generation 64

65 FIR CIC clean-up Filters The sinc^ near-ero frequency response is often compensated for using an FIR filter H FIR e sin w sin w K Estimating useful narrowband bandwidth: Find attenuation required using 3 or 4 stage CIC Chap_3.m provides attenuation and alias curves WAG based on 3 stage single sideband (LPF), -60 db atten., approx. /0 th of the output sample rate (/0*) 65

66 CIC Clean-up Filter See AlteraCICFIR.m tap FIR filter 0 0 Design FIR 7 Design FIR x x

67 Digital Logic Devices and Notes Altera Xilinx Texas Instruments 67

68 Application Areas Digital up and down converters Sigma-Delta Analog-to-Digital Converters Whenever high clock rate, low bit precision data is available for processing. Note both decimation and interpolation 68

69 Additional References Hogenauer, E.;, "An economical class of digital filters for decimation and interpolation," Acoustics, Speech and Signal Processing, IEEE Transactions on, vol.9, no., pp. 55-6, Apr atthew P. Donadio, CIC Filter Introduction, web article, 8 July