ece 6560 multirate signal processing chapter 11bazuinb/ece6560/chap_11.pdf · ece 6560 notes and...

ECE 6560Multirate Signal Processing

Chapter 11

Dr. Bradley J. BazuinWestern Michigan University

College of Engineering and Applied SciencesDepartment of Electrical and Computer Engineering

1903 W. Michigan Ave.Kalamazoo MI, 49008-5329

ECE 6560 Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-146511-2.

2

Chapter 11: Cascade IntegratorComb Filters

11.1 A Multiply Free Filter 32611.2 Binary Integers and Overflow 33311.3 Multistage CIC 33611.4 Hogenauer Filter 341

11.4.1 Accumulator Bit Width 34211.4.2 Pruning Accumulator Width 344

11.5 CIC Interpolator Example 35611.6 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.29, no.2, pp. 155- 162, Apr 1981.http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1163535


3

A Multiplier 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 13-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.


4

Frequency Response and Implementation (infinite sum)

1

0exp1

M

n

jw nwieH

Mnn

jw nwinwieH exp1exp10

0

exp1exp1n

jw nwiMwieH

0'0

'exp1exp1nn

jw MnwinwieH

'expexp1exp10'0

nwiMwinwieHnn

jw


5

Frequency Response and Implementation (infinite sum)

21sin2

2sin2

21exp

2exp

wi

Mwi

wi

MwieH jw

21sinc

2sinc

21exp

21sin

2sin

21exp

w

MwMMwi

w

MwMwieH jw

wiMwieH jw

exp11exp1

Mw

2Nulls at:

0

exp1exp1n

jw nwiMwieH


6

Frequency Response and Implementation (textbook)

21sin

2sin

M

H

1

0

M

kknxny

Nk

2

First Null at:

M 2

Note: The frequency response approximates a circular sinc function

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 narrower and more

sloped.


7

4

4

21sinc

2sinc

w

MwMeH jw

Solved one problem …But this is a lot of memory and summing.


8

Boxcar Equivalent: Comb-Integrator (Finding another way with math equivalents)

1

0

M

k

knxny

MxknxnxknxnyM

k

M

k

1

0

1

01

11 nyMxnxMxnynxny

1

0

11M

k

knxnyLet and

An iterative update equation can be defined as

The “Boxcar” implementation as comb and integrator filters!


9

Comb-Integrator Response

1 nyMxnxny

zYzzXzzXzY M 1

MzzXzzY 11 1

111

zz

zXzY M

222

222

11

wjwjwj

wMjwMjwMj

jw

jwM

jw

jw

eee

eee

ee

eXeY

2sin

2sin

2sin22sin2

11 2

1

2

2

w

wMe

wje

wMje

ee

eXeY Mwj

wj

wMj

jw

jwM

jw

jw

Equivalent Spectral Domain

MATLAB Response

• Function sin/sin or sinc/sinc• If the signal bandwidth is very narrow, it works.

– What would happen if you decimated by M ? – The sinc/sin zeros alias/align to the 0 frequency….


10

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-120

-100

-80

-60

-40

-20

0

Frequency

Pow

er (d

B)

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency

Pow

er (d

B)

Band edge power = -10.4231 dB

1st NullPassband

LOOKING AT THE COMB AND INTEGRATOR


11


12

Comb Only Response Mxnxncomb

zXzzXZComb M

jwM

jw

jw

eeX

eComb 1

2sin2 2 wMejeX

eComb wMjjw

jw

MzzXZComb 1

222wMjwMjwMj

jw

jw

eeeeX

eComb

M periodic zeros – comb teeth?• Null specific frequencies


13

Comb Filter

MMk

M

kw 122

12

• A marginal filter for signal– Narrow “shaped” passbands

– Nulls at k = integer

kwMforwM2,2sin0

22,2sin1 kwMforwM

Mkw 2

Peak and null locations


14

Integrator Only Response

0n

nxncint

0n

nzzXzCint

jwjw

jw

eeXeCint

11

2sin22sin2

22

2 we

we

jeXeCint

wj

wjjw

jw

222

1wjwjwjjw

jw

eeeeXeCint

For 0 <= w <= 2

“” at w = 0 and w = 2• Sinc function “adjustment”

Magnitude “1” at w = ±

11

zzXzCint

Combining the Comb and Integrator

• Combining Spectral Responses– Infinity and zero combine to be a constant, otherwise comb filter

elements persist with periodicity M.


15

2sin2 2 wMejeX

eComb wMjjw

jw

2sin2 2 we

jeXeCint

wjjw

jw

2sin

2sin

2sin22sin2

11 2

1

2

2

w

wMe

wje

wMje

ee

eXeY Mwj

wj

wMj

jw

jwM

jw

jw


16

CIC Filter Structure

• Only two summation (adds) are required.• Original “Boxcar” required M-1 summations

– An M element memory is still required (but we haven’t decimated yet)

MATLAB Response

• Function sin/sin or sinc/sinc prior to decimation by M.


17

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-120

-100

-80

-60

-40

-20

0

Frequency

Pow

er (d

B)


18

CIC Alternate Derivation

• Recognition of mathematical sequences …

zXzzXzzXzzXZY M 121

1

0

M

k

knxny

1

0

M

k

kzzH

1210 MzzzzzH

1

1111

111

zz

zzzzH

M

M

M


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– Mathematically this has problems due to the implied canceling of

the zero and pole at w=0! (See the comb plot)

1111z

zzH M


20

Architecture Impulse Responses(Identical output, different “internal” results)


21

Impulse Response

• Comb-Integrator– Filter operates as expected

• Integrator-Comb– Filter operates, but continues to recalculate input even after the

filter has stabilized at a steady-state value.– Internal node values do not return to zero with a zero 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 stabilize 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”!


24

• 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

1,0;21

0

i

b

i

ii aaN

00

11

22

11 2222

aaaaN bb

bb

1,0;222

0

11

i

b

i

ii

bb aaaN

00

11

22

11 2222

aaaaN bb

bb


25

Two’s Complement Overflow

Figure 11.7 presents the overflow behavior of a 2’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 2’s-complement accumulator leads to the correct answer as long as the accumulator is wide enough to hold the correct answer.


26

CIC Filter Example

• CIC with M=4 & x(n)=1– The integrator-comb

architecture can function!– Note the structure … if

decimation were to occur.


27

Multistage (Cascaded) IC

• If one works, why not cascade a number of IC stages?


28

• For one stage

• For two stages

• For K stages

Multistage (Cascaded) IC Response

111

zzzH

M

2

111

zzzH

M

KM

zzzH

111

2sin2sin

w

wMeH jw

2

2sin2sin

w

wMeH jw

K

jw

w

wMeH

2sin2sin

ECE 6560 Notes and figures are based on or taken from materials in the course textbook: fredric j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall PTR, 2004. ISBN 0-13-

146511-2.

29

Impulse Responses M=10


146511-2.

30

Frequency Response


31

Frequency Response Comments

• The rect to sinc response only provided –13 dB attenuation of the first sidelobe

• Each successive stage is provides an additional –13 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!


32

Frequency Response Detail


33

CIC Suppression of Example


34

CIC Null Suppression

• Matlab Chap11_2.m and Chap11_3.m

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-120

-100

-80

-60

-40

-20

0

Frequency

Pow

er (d

B)

Multistage CIC

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025-120

-100

-80

-60

-40

-20

0

Frequency

Pow

er (d

B)

Band edge power = -62.244 dB

1st NullPassband


35

CIC Suppression Zones

• This capability is similar to having don’t care filter zones in Chapter 7.

CIC Suppression Zones


36

• For a narrow bandwidth replicated signals, the suppression zones provide useful filtering!

Suppression Zones

• For what operations do with discover “replicated zone” activities– Most obvious – interpolation … the zones make good filters!– Useful for interpolate by M.

– What about aliasing zones when decimation is applied. – The zones would zero any signals that would alias to the narrow

bandwidth at the origin …. again a useful filter!– Useful for decimate by M.


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?


39

Hogenauer Filter

• Using a CIC filter as the filter for interpolation-filter or filter-decimation

Interpolation-Filter Filter-Decimation


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 M, the nulls in the spectrum will eliminate signals (interpolated or aliased) about a specified narrow bandwidth at the location of the nulls.


41

Multistage Filter Decimation Example

• Basic Cascade

• Reorder

• Noble Identity


42

One Major Concern – Bit Widths

• The comb stages must have sufficient input bit sizes 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

11 MzHGain

KK MzHGain

Kdatafilter Mceilbb 2log

MKceilbb datafilter 2log


43

Example: K=2, M=20, bdata=7 400202 zHGain K

169732.42720log27 2 ceilceilbfilter

• Time series output using a maximum cosine value


44

If too few bits are allowed …

• Non-sinusoidal output!


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.


46

Up-Sampling CICFigure 11.17 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.

kMkMk

zzz

zXzY

1

4

111


47

CIC-Interpolator: Integrator Inputs

Hogen16b example:

16-tap, 3 stage


48

Maximum Signal Gain

• Based on the impulse responses nysignnx kk


49

Resulting Maximum Integrator Responses

• Note that each integrator is maximized based on a different input signal.

• To define the bit widths needed, use the maximum for the stage!– Comb filters +1 bit per stage (adder)– Integrators bits based on max gain (8, 11, 14, 18)


50

Maximum Integrator Output


51

Maximum Response per Stage


52

Bit Growth

8

11


53

Down-Sampling CICFigure 11.24 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.

kM

kM

k

zzz

zXzY

4

1 111


54

CIC-Decimator Integrator Outputs


55

Noise-Power Gain

• Example Noise Gain (referred to output)

k

N

n

GNngkkkkOut

21

0

222)(


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,


57

Bit Decrease


58

CIC Interpolator Example

• Polyphase interpolate-filter by a factor of 5– Shape the baseband input

• CIC up convert by a factor of 16– Hogen16b.m and hogan16c.m

• Resulting interpolation rate is 5 x 16 = 80!


146511-2.

59

Shaping Filter


146511-2.

60

Shaping Pulse Responses


146511-2.

61

Composite Output Spectrum

MATLAB Simulation

• See Chap11_8.m: Cascaded Nyquist shape interpolator (x5) and 3 stage CIC interpolator (x16). Total interpolation (x80)– fsymbol = 1;– fsample1 = 5;– fsample2 = fsample1 * 16;– 3-stage CIC interpolator

– Input: Impulse response or ASK symbols


62


63

Author’s Examples

• Hogen16a• Hogen16b• Hogen16c• Hogen50 /Chap11_5

– Add_2

Additional Simulation

• Chap11_6.m Modified hogen16a• Chap11_7.m hogen16a decimator• Chap11_8.m Section 11.5 example interpolator

• Notes on Nyquist filter generation


64

FIR CIC clean-up Filters

• The sinc^M near-zero frequency response is often compensated for using an FIR filter

• Estimating useful narrowband bandwidth:– Find attenuation required using 3 or 4 stage CIC

Chap11_3.m provides attenuation and alias curves• WAG based on 3 stage single sideband (LPF), -60 dB atten.,

approx. 1/10th of the output sample rate (1/10*M)


65

K

jwFIR w

wMeH

2sin2sin

CIC Clean-up Filter

• See AlteraCICFIR.m– 111 tap FIR filter


66

0 2 4 6 8 10 12

x 106

-160

-140

-120

-100

-80

-60

-40

-20

0

20

DesignFIR2

0 1 2 3 4 5

x 106

-3

-2

-1

0

1

2

3

4

5

6

7

DesignFIR2

Digital Logic Devices and Notes

• Altera– http://www.altera.com/literature/an/an455.pdf

• Xilinx– https://www.xilinx.com/products/intellectual-property/cic_compiler.html

– Designing Efficient Digital Up and Down Converters for Narrowband Systems

• https://www.xilinx.com/support/documentation/application_notes/xapp1113.pdf


67

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

Additional References

• Hogenauer, E.; , "An economical class of digital filters for decimation and interpolation," Acoustics, Speech and Signal Processing, IEEE Transactions on , vol.29, no.2, pp. 155- 162, Apr 1981.http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1163535

• Matthew P. Donadio, CIC Filter Introduction, web article, 18 July 2000.http://dspguru.com/dsp/tutorials/cic-filter-introductionhttp://dspguru.com/sites/dspguru/files/cic.pdf


69

ece 6560 multirate signal processing chapter 11bazuinb/ece6560/chap_11.pdf · ece 6560 notes and...

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