ECE 6560 Multirate Signal Processing Chapter 4bazuinb/ECE6560/Chap_04.pdf · ECE 6560 Notes and...

ECE 6560Multirate Signal Processing

Chapter 4

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 4: Useful Classes of Filters

4.1 Nyquist Filter and Square-Root Nyquist Filter 824.2 The Communication Path 864.3 The Sampled Cosine Taper 894.3.1 Root-raised Cosine Side-lobe Levels 914.3.2 Improving the Stop-band Attenuation 924.4 Half-band Filters 97


3

A Communication System

• Digital communication systems transmit a sequence of symbols. Due to filtering the receiver filter response from one symbol may overlaps that of another symbol, resulting in intersymbol interference (ISI). – ISI is a coherent error term that directly degrades our ability to

resolve the current symbol.

• The goal is to define digital filters that, when sampled at the appropriate time, will zero any ISI. – if not correctly sampled, there will be ISI.– See: http://complextoreal.com/ by Charan Langton, Tutorial #14

Demonstration of MPSK and MQAM with Square Root Nyquist Simulations• Advanced Digital Communication Tool Demo

– BER_Test_NyquistFilter– BER_Test_Time


4

5

Model of a CW communication system with noise: Figure 10.1-1

CW Communication with Noise

ttf2cosLtAtx cc

tnttf2cosLtAtv c

thtnttfLtAteD Rc

2cosPr

ttf2costAtx c

ECE 6560Notes 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.

6

Digital Formatting and Transmission

ECE 6560Notes 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.


7

Filtering Comm. Symbols

• General Filter Concept

The scaling terms d(n), are selected from a small finite alphabet such as for BPSK {-1, +1} or for ASK {-1,-1/3, +1/3, +1} in accord with a specified mapping scheme between input symbol (bits) and output levels.

The signal s(t) is sampled at equally spaced time increments identified by a timing recovery process in the receiver to obtain output samples as shown in Eqn. (4.2).

n

Tnthndts

n

TnTmhndTms

8A. Bruce Carlson, P.B. Crilly, Communication Systems, 5th ed.,

McGraw-Hill, 2010. ISBN: 978-0-07-338040-7.

(a) signal plus noise (b) S/H output (c) comparator output: Figure 11.2-2

Regeneration of a unipolar signal

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.


9

Filter Concept

We can partition this sum as shown in Eqn. (4.3), to emphasize the desired and the undesired components of the measurement. Here the desired component is d(m) and the undesired component is the remainder of the sum which if non-zero, is the ISI.

How do we eliminate the intersymbol interference (ISI) ?

Let the time/sample representation of the filter be.

mn

TnTmhndhmdTms 0

0,10,0

nn

Tnh “Perfect time sampling is implied”

Possible Time-Based Filters


10

• They could be multiple symbols in length if they are zero at all integer values except n = 0

• Do we already know of a filter with this characteristic? (What about a time domain Sinc ?!)

“Function/filter must be zero at all integer values except n = 0”

“The shape of the filter must have a

reasonable frequency bandwidth, too”


11

Sinc as a Zero ISI Filter

Considering the spectral and time domain requirements, we can also use

Tt

Tt

th

22

22sin

n

n

TTn

TTn

Tnh

sin

22

22sin

Sinc Function and “Reconstruction”

• The “convolution” of the sinc function with sampled time waveform “impulse samples” is how perfect band-limited signal reconstruction is performed. – The continuous time sinc is the time-domain transform of the

perfect frequency-domain “brick-wall low pass filter”

• For symbols, we only need to “reconstruct” the symbol value without ISI at one time instant during the symbol period.– Nominally select the center of the symbol.– The “reconstructed” continuous time signal need not look like the

original symbol waveform (they have significantly different frequency spectra and bandwidth!)


12

Rough Example

• See SincEye.m amd SincEyev2.m– Each Symbol represented by a multi-cycle sinc function– The nulls of the sinc function occur at the “optimal” symbol

sample point. All other sample points would be required to sum the signals levels from the other symbols (symbol interference).

– Therefore, to limit ISI, you must 1. Properly filter2. Properly (perfectly) sample in time


13

Optimal Filter for Pulse Detection (1)

• If we want to detect a transmitted pulse with maximum SNR, the following applies

ECE 6560 14

EQNoise

Signal

BNtsE

PP

SNR

0

2

tnts tnts oo filtered to

0

dtntshtnts oo

0

2

2

0

21 dtthN

dtshE

SNR

o

out

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.

Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.


• Applying Schwartz’s Inequality to the output SNR

• The upper bounds on the SNR may be defined as

• But we can also define a condition for “equality”

ECE 6560 15

0

2

0

2

2

0

dsdhdsh

0

2

0

2

0

2

0

2

2

21

dtsEN

dtthN

dtsEdhSNR

oo

out




• For equality to exist

• A possible solution is

• This is an “optimal inverse-time filter”– The filter is the inverse time response of the transmitted pulse s(t)!

ECE 6560 16

0

2

0

2

2

0

dtsdhdtsh

utsKh




• Continuing for completeness– The desired impulse response is simply the time inverse of the signal

waveform at time t, a fixed moment chosen for optimality. If this is done, the maximum filter power (with K=1) can be computed as

• And the maximum output SNR becomes

• The filter is commonly called a “Matched Filter”

ECE 6560 17

tdsdtsdht

2

0

2

0

2

tN

SNRo

out 2max




18

An Approach to Generating Filters

1. Defined the desired/required/stuck-with symbol spectrum or “time pulse” with a finite duration (less than or equal to the symbol period).

2. Multiply by the sinc in the time domain– Convolve in the frequency domain– Infinite time /non-causal nature still a problem– This enforces the h(nT) requirement!

3. Apply a time domain window after spectral filter design– Modify passband and stopband ripple and edges as needed


19

Spectral Convolutions

Note: frequency domain shown as two-sided spectrum widths

Zero Time Requirements

Window Function

Spectral Convolution /T for 0.1<<0.5

Infinite Sinc for ISI

Symbol Time Window, finite time, small BW penalty

Windowed, zero ISI filter


20

Using Previously Defined Windows

• Start with the time-domain sinc function– Determine the filter length based on the number of sinc cycles to

be maintained (null-to-null samples from +/-1st, +/-2nd, +/-3rd, etc.– The Fourier transform has a sin(n)/sin() shape with frequency

periodicity based on the number of sinc cycles.

• Generate a window of the same number of samples– Multiply in time domain, convolve in frequency domain.– Removes the Gibbs phenomenon peaks and reduce the passband

ripple.

• Is there a preferred window/filter? (Yes, raised Cosine)

Web References

• Wikipedia– InterSymbol Interference (ISI)

• http://en.wikipedia.org/wiki/Intersymbol_interference– Nyquist ISI Criterion

• http://en.wikipedia.org/wiki/Nyquist_ISI_criterion– Inter Symbol Interference (ISI) and Raised cosine filtering

• http://complextoreal.com/wp-content/uploads/2013/01/isi.pdf• From C. Langton “Complex to Real” web site

– A windowed sinc function will be used for ISI• The window often applied is the raised cosine

ECE 6560 21Notes 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.


22

Nyquist Filtering with Raised Cosine

• The Nyquist pulse is the wave shape required to communicate over band-limited channels with no ISI.– It is generated as a raised cosine frequency spectrum window

• Even symmetric spectral window.• Finite frequency width that is a fraction of the perfect

reconstruction width. (i.e. /T)– Preference to limit the time response to a length 4T/– Truncated window (window length) and infinite sinc

• With convolution in the frequency domain, the spectrum becomes a width of (1+)/T


23

Spectral Convolutions (1)

• The term is called the roll-off factor and is typically on the order of 0.1 to 0.5 with many systems using values of = 0.2. (Typically: 0< <1)– The transition bandwidth caused by the convolution is seen to

exhibit odd symmetry about the half amplitude point of the original rectangular spectrum.

– This is a desired consequence of requiring even symmetry for the convolving spectral mass function. When the windowed signal is sampled at the symbol rate 1/T Hz, the spectral component residing beyond the 1/T bandwidth folds about the frequency ±1/2T into the original bandwidth.

– This folded spectral component supplies the additional amplitude required to bring the spectrum to the constant amplitude of H(f).


24

Spectral Convolutions (2)

• We also note that the significant amplitude of the windowed wave shape is confined to an interval of approximate width 4T/ so that a filter with = 0.2 spans approximately 20T, or 20 symbols in duration! – We can elect to simply truncate the windowed impulse response to

obtain a finite support filter, and often choose the truncation points at ± 2T/ or 10 symbols.

– A second window, a rectangle, performs this truncation. The result of this second windowing operation is a second spectral convolution with its transform. This second convolution induces pass-band ripple and out-of-band side lobes in the spectrum of the finite support Nyquist filter. (Nothing is perfect ….)


25

Symbol Periods in Communications“Let’s do the math”

A communication system can be modeled most simply by the signal flow shown in Figure 4.4. Here d(n) represents the sequence of symbol amplitudes presented at symbol rate to the shaping filter h1(t).

Perfect time sampling provides the detected symbol output …. the equivalent of filter-decimation!


26

Symbol Periods Math (1)

Filter received signal

Tmthmdtsm

1

tntstr

thtrty 2

dhtntsdhtrty 22

dhtndhTmthmdty

m221

tndhTmthmdtym

221


27


Equivalent filter of the received signal

tndhTmthmdtym

221

dhthtg 21

Define the convolution of the transmitter and receiver filters

tnTmtgmdtym

2

The received signal is then


28


Minimally sampling the symbol outputTnt

TnnTmngmdTnym

2

TnnTmngmdgndndTnynm

20ˆ

Expanding


29

Discrete Samples at the Symbol RateInterpretation

1. Desired Signal (convolved filter, prefer a matched filter)2. Band limited Noise (n2(t)) filtered by h2(t)

(typically filtered white noise, for power use BWeqn

3. Combined ISI (remove with Nyquist filter g(t))

We need g(nT) to be a Nyquist filter to remove ISI !!

TnnTmngmdgndTnynm

20

delayTwjNyquist ewHwHwHwG 21

1 23

Transmit and Receive Filters

• We want the result to be a Nyquist Filter with time delay

• Using a matched filter for H1 and H2 should provide maximum outputs. Make a Square-root Nyquist filter!


30

delayNyquist TwjfHwHwH exp21


31

One Approach:Square Root Nyquist Concept

To maximize the signal-to-noise (SNR) in (4.10), the receiver filter must be matched to the transmitter-shaping filter. The matched filter is a time-reversed and delayed version of the shaping filter, which is described in the frequency domain as shown in (4.11). Let

delayTwjwHconjwH exp12

delayNyquistdelay TwjwHTwjwHwHwH expexp2121

wHwH Nyquist21

wHwH Nyquist1


32

Square Root Nyquist


33

Second Approach:Square Root Nyquist Concept

Use the equivalent of a ZOH output ….

symbolsymbol Ttrect

Tth 1

1

delayNyquistdelay TfjfHTfjfHfH 2exp2exp21

symbolTffH sinc1

delay

oncompensati

Nyquist TfjfH

fHfH 2exp

_12

Note that the Nyquist filter is formed from a sinc basis, therefore the nulls appear at the same locations as the Nyquist filter! Let,

Nyquist and Square Root Nyquist

• Well defined time and spectral responses …

• Wikipedia Raised-Cosine Filter– http://en.wikipedia.org/wiki/Raised-cosine_filter


34


35

The most common spectral mass selected for communication systems is the half cosine of width *fSYM. The half cosine convolved with the spectral rectangle forms the spectrum known as the cosine-tapered Nyquist pulse with roll-off .

The description of this band-limited spectrum normalized to unity pass-band gain is presented in (4.14).

Nyquist Filter

221

cossintftf

tftf

fthSym

Sym

Sym

SymSymNyq

Sym

SymSym

Sym

Nyq

ww

for

ww

forw

w

ww

for

wH

10

1112

cos15.0

11


36

Discrete Time Filter

• Let fsample = M*fsymbol so that fsymbol*t is replaced by fsymbol*n/(M * fsymbol) or n/M.

• It is common to operate the filter at M= 4 or 8 samples per symbol

2

21

cossin

SymSym

SymSym

SymSym

SymSym

Symsample

Nyq

fMnf

fMnf

fMnf

fMnf

ff

nh

2

21

cossin1

MnMn

MnMn

MnhNyq


37


• The filter described in Eqn. (4.16) has a two-sided bandwidth that is approximately 1/Mth of the sample rate. A digital filter exhibits a processing gain proportional to the ratio of input sample rate to output bandwidth, in this case a factor of M. The 1/M scale factor in Eqn. (4.16) cancels this processing gain to obtain unity gain. When the filter is used for shaping and up sampling, as it is at the transmitter, we remove the 1/M scale factor since we want the impulse response to have unity peak value rather than unity processing gain.

2

21

cossin1

MnMn

MnMn

MnhNyq


38

Square Root Nyquist Filter• The square root of the cosine-tapered Nyquist filter results in a

quarter cycle cosine tapered filter. This description is normally confined to square-root raised cosine or root raised cosine Nyquist filter. The description of this band-limited spectrum normalized to unity pass-band gain is shown in (4.17).

Sym

SymSym

Sym

NyqSqrt

ww

for

ww

forw

w

ww

for

wH

10

1114

cos

11

tftf

tftftffth

SymSym

SymSymSymSymNyqSqrt

241

1sin1cos4

Textbook sign error!


39


• Let fsample = M*fsymbol so that fsymbol*t is replaced by fsymbol*n/(M * fsymbol) or n/M.

SymSym

SymSym

SymSym

SymSym

SymSym

SymSample

NyqSqrt

fMnf

fMnf

fMnf

fMnf

fMnf

ff

nh

2

41

1sin1cos4

Mn

Mn

Mn

Mn

Mn

Mnh NyqSqrt

2

41

1sin1cos41 Textbook errors!

Sign and extra n.

Generating Filter in Matlab

• See Chap4_1.m introduce nyquistfilt.m• See Chap4_2.m introduce sqnyquistfilt2.m• See Chap4_3.m nyquistfilt.m vs. firrcos.m

(vary alpha & length)• See Chap4_4.m nyquistfilt.m vs. firrcos.m vs. rcosfir.m

(vary alpha & fixed length)• See Chap4_5.m sqnyquistfilt2.m vs. square root firrcos.m

vs. square root rcosfir.m(vary alpha & fixed length)


40

Contemplating the passed variables

• See Chap4_4.m for an example• nyquistfilt

– Alpha, fsample/fsymbol (samples per symbol), 2*k symbols is length of the filter (+1 so it is odd length)

• Firrcos– N+1 filter length, alpha, fsample frequency, fsymbol/2 cutoff

frequency

• Rcosfir– R=Alpha rolloff, T=1/fsample, rate = fsample/fsymbol (samples

per symbol), 2*k symbol length filter (+1 so it is odd)


41


42

Root-raised Cosine Side-lobe LevelsWe commented earlier that when we implement the SQRT Nyquistfilter, we actually apply two windows; the first window is a smooth continuous function used to control the transition bandwidth and the second is a rectangle used to limit the impulse response to a finite duration.

This second windowing forces side lobes in the spectrum of the SQRT Nyquist filter. These side lobes are quite high, on the order of 24 to 46 dB below the pass band gain depending on roll-off factor and the length of the filter in number of symbols.


43

Root-raised Cosine Side-lobe Levels

2*k*M filter+1 length (fsample=M*fsymbol and k symbols)

Chap4_4.m gets different values?!


44

Response Problems

• The reason for the poor side-lobe response is the discontinuous first derivative at the boundary between the half-cosine transition edge and the start of the stop band. Consequently the envelope of the time function falls off, as seen in Eqn. (4.18), as 1/t^2 enabling a significant time discontinuity when the rectangle window is applied to the filter impulse response.

• “In retrospect, the cosine tapered Nyquist pulse was a poor choice for the shaping and matched filter in communication systems.” p. 91

tftf

tftftffth

SymSym

SymSymSymSymNyqSqrt

241

1sin1cos4


45

Other Windows

• Attempting to control the spectral side lobes by only applying other windows to the weights of the prototype (sinc) filter results in significant increase in the ISI levels at the receiver output.

• This is illustrated in Figure 4.7, which illustrates the effect on spectral side-lobes and ISI levels as a result of applying windows to the prototype impulse response.

• The increase in ISI is traced to the shift of the filter’s 3-dB point away from the nominal band edge. The requirement for zero ISI at the output of the matched filter requires that the shaping and matched filters each exhibit 3-dB attenuation at the filter band edge, half the symbol rate.


46

Other Windows


47

Matlab code example

• Nyq_2aaTest.m– Special f. harris functions

• NyquistTestv0.m– Compare the number of symbols used (increased filter length)

• NyquistTestv1.m– Validate Dr. Bazuin’s filter routines nyquistfilt.m as compared to

firrcos.m

• NyquistTestv2.m– Validate f. harris nyq_4 filter routines (nyq_fharris) as compared to

firrcos.m


48

Matlab code example (cont)

• NyquistTestv3.m– Observing firrcos.m

• NyquistTestv4.m– Testing nyquistfilt.m and nyquistfilt_even.m

• NyquistTest.m– Testing nyquistfilt.m and sqnyquistfilt2.m

Improved Nyquist Filter

• Software Defined Radio (SDR) Forum '05 Papers, November 14-18, 2005 - Hyatt Regency - Orange County, California .

• An Improved Square-Root Nyquist Shaping Filter– harris f., Chris Dick, S. Seshagir, Karl Moerder; San Diego State

University, Xilinx, Broadband Innovations– Still available, but you will need to do a web search …– This paper appears to be the original source for section 4.3.2.


49

Recreating the Textbook

• See ISI-Sq_Nyq_2_Test.m– Note: hh length is set by NN (odd), firrcos must be odd length

• Nyq_2aaTest.m compares nyq_fharris.m to new harris!


50

MATLAB Comm Toolbox Demo

• See RCosTestDemo.m– Raised cosine example

Direct FIR filtering– Square Root Raised Cosine Transmit and Receive

Direct FIR filtering– Polyphase Square Root Raised Cosine Transmit and Receive

MATLAB Toolbox Implementation

– Cost analysis using MATLAB Toolbox


51

Generating a Signal

• ASKGen.m– Self contained generator for creating ASK symbols based

• ASK_TSG_GEN.m– Routine to plot generated ASK and then show “received” eye diagrams based on


52

0 1000 2000 3000 4000 5000 6000-2

0

2Test Data

FilteredInterp. Symbol

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-100

-50

0

50Test Data Spectrum

FilteredInterp. Symbol

ASK Test Signals

• The eye diagrams following square root Nyquist receiver filters, firrcos and harris’s Nyquist filter.

• This is also an example of symbol interpolation into a sample waveform and filtering …


53

Shifting Gears

• Introduce the half-band filter– Based on rect in the frequency domain– Takes advantage of unique real-symmetric type 1 FIR features

– Half-bands will be revisited in Chapter 8


54

DSP Class Review(Performed in previous lecture)

• A review of FIR Filters from ECE4550/5550Digital Signal Processing

• Conversions Examples for Type 1 Low Pass Filters– LP to HP Delay Complementary Translation– LP to HP Frequency Translated Filter– LP to FT HP to DC LP Translation


55

Filter Type # Coef., Symmetry CommentsType 1 Filter: Odd, Symmetric It can do all filters.Type 2 Filter: Even, Symmetric It can not be a highpass filter.Type 3 Filter: Odd, Anti-Symmetric It can only be a bandpass filter.Type 4 Filter: Even, Anti-Symmetric It can not be a lowpass filter.

MATLAB Examples

• See firPMCompFilter.m• See firPMTransFilter.m• See firPMTransComp.m


56


57

Half Band Filters

• fs/4 centered signal spectral band edge (i.e. ½ voltage)

Half-Band Filters

• Given the desired responses based on rectangular frequency domain low pass filters and the resulting time domain sinc filter …

• Can we select appropriate bandwidths that zero out some (or even many) of the coefficients? – Simplify filter and reduce computations required.


58


59

Rectangular Low Pass Filter

12

rectf

ffH

tffth 11 2sinc2

tf

tffth

1

11 2

2sin2

tf

tf

fth

222

222sin

21

1

1

11

11

21

222:

ft

tfzeroFirst

zerost

zerost


60

Sampled Impulse Response

ssss fnt

tftf

ff

fnt

fth

1

11

22sin21

s

s

ss

fnff

nf

ff

fnh

1

1

1

2

2sin2 1

12 sf

ffor

1

11 sin

n

nnhfnh s

s


61

Half Band Sinc

tf

tf

fth

222

222sin

21

1

1

2support

1

Ff

sfnt

n

fF

nf

F

fF

nh

s

s

s support

support

support

sin


62

Half Band Sinc (2)

Note: for n even (except 0) h(n)=0!

2

2sin

21

n

nnh

2supportsfF

Only one-half the coefficients are needed

The coefficients are symmetrical, thus only one quarter are unique

An efficient implementation can use N/4 multipliers for an N+1 length filter!

41sff or

Then


63

Half Band Smoothing Filter

• Zero Coefficients remain, others modified/windowed• Convolution in the spectral domain about fs/4

nwnhnh bandhalfLOW


64

Generating a High-Pass Filter

• Modulate the complex spectrum by fs/2– Frequency Translated Filter

tfjtftfi sss

22sin

22cos

22exp

sfnt

njnfnfi

s

s

sincos2

2exp

ns

s nfnfi 1cos

22exp

Odd coefficients are negated. h(0)

not negated.


65

High Pass Filter

For h(0) = 1/2,Frequency shift and delay

complementary translation are identical.


66

Note Common Coefficients

• Defining two prototype filter (even and odd)


67

This Leads to our first polyphase structure

• This is also referred to as a Quadrature Mirror Filter– Note the spectral mirroring about the fs/4 point


68

Non-causal Discrete Implementation

• Since

Half-Band Filter Coefficients

• From the definition of a half-band filter,

• Define even and odd time samples

• Make the Filter Causal (for a 2N+1 tap filter)


69

Note: for n even (except 0) h(n)=0!

2

2sin

21

n

nnh

nh 2 12nh

Ncausal znhnh


70

Causal Discrete Implementation

MATLAB

• See HB_TEST.m– hbfilt_gen.m (sinc or remez)– hbfilt_pmgen.m (forced “half-band” symmetry generation on firpm

followed by forced zeroing of even non-zero coef.)

• WavTest.m– WavFiles– waveplay2.m– Audio signal for processing inside of MATLAB


71


72

Chapter Problems

• Generating the Nyquist and Square Root Nyquist Filters– The definitions are available but

• The number of taps has not been defined, N=f(M, k, )• There are zero valued elements that must be correctly analyzed in the

denominators. (can use sinc example for near singularities)

• Alternate scripts are available for you to use.– If you have access to the DSP toolbox, compare it!

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