Makalah Elektro - (Digital) Signal Processing

Signal Processing, 2004/05 — 1. Overview S. Weiss

(Digital) Signal Processing — Overview

• In the first part, you have studied a good deal of background theory;

• we want to explore some more (particularly digital) signal processing theory,applications, and architectures;

• digital signal processing (DSP) is based on theory that goes back more than 200years to Norbert Gauss and others;

• DSP has seen a revolution due to the arrival of inexpensive computational devicesin the 1990;

• DSP has penetrated almost every aspect of life: Texas Instruments claims thatevery 10 minutes you use one of their dedicated digital signal processors.

1


DSP Applications

• Multimedia: compression and coding algorithms for audio, video, and still images;acoustic echo control for teleconferencing;

• Audio: audio surround sound, MP3, seamless audio interfaces;

• Communications: equalisation, detection, smart antenna processing;

• Biomedical: signal detection; source separation for ECG or EEG; digital hearingaids;

• Defense: radar/sonar; detection and guidance systems;

• Automotive: GPS navigation; engine management;

2


DSP Perspective: Software Defined Radio

• Reliability, repeatability and programmability of DSP has widely replaced analoguesystems in consumer and industrial markets;

• an ambitious future system relying on DSP is a software defined radio:

at RF or IF

ADC

DAC

DSP

etcdatavideoaudio

sampling

Sampling at an intermediate or even radio frequency (IF or RF) permits flexibilityof the modulation stage and use of advanced DSP techniques for synchronisation,equalisation, carrier offset recovery etc.

3


Notation — Time Domain

• Continuous time signal: x(t) with t ∈ R; example: analogue speech signal or anyother voltage signal;

• discrete time signal x[n] withn ∈ Z; example: daily stockprice, video signal:

• vector quantity: x, if timedependent x[n] or xn

• matrix quantity: Tn+1

n+2

+3n

+4n

ff

ff

n

ftime

space

4


Notation — Transform Domain

• Fourier transform:

x(t) —• X(jω) X(jω) =

∞∫−∞

x(t)e−jωt dt (1)

x[n] —• X(ejΩ) X(ejΩ) =∞∑

n=−∞x[n]e−jΩn (2)

with angular frequency ω = 2πf and normalised angular frequency Ω = 2πω/ωs

for sampling at ωs.

• note that X(ejΩ) is periodic with 2π.

• z-transform: H(z) •— h[n] such that H(z) =∞∑

n=−∞h[n] z−n

5


Notation — Stochastic Signals

• A stochastic signal x[n] can be characterised by its probability density function(PDF);

• often we are interested in its autocorrelation function

rxx[τ ] = Ex[n]x∗[n− τ ] (3)

with lag τ , complex conjugation operator ·∗ and expectation operator E·;

• power of x[n] is given by rxx[0]; if x[n] is zero mean, then the variance σ2xx = rxx[0];

• power spectral density (PSD) of x[n]:

Rxx(ejΩ) •— rxx[τ ] such that Rxx(ejΩ) =∞∑

n=−∞rxx[τ ]e−jΩτ (4)

6

Signal Processing, 2004/05 — 2. Conversion S. Weiss

Electrical (Voltage) Signals

• A signal is a variation over time of a measurable quanity that conveys informationof some form;

• a sensor (actuator) can convert a real world signal to (from) a convenient voltagesignal:

0 1 2 3 4 5 6 7 8 9 10

−2

−1

0

1

2

time t / [msec]

ampl

itude

s(t

) / [

V]

(ts )

wavepressure

signalelectrical analogue

transducer

• other conversion examples: pressure — piezoelement; light — photosensor;vibration — accelerometer; electro-magnetic — antenna.

7


Amplification and Noise

• Many real world signals yield small amplitudes (e.g. EEG order of 10−6v), andamplification is required prior to further processing;

• generally such signals are corrupted by noise; we assume an additive noise model:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

time t / [s]

v(t)

n]v

[n]y[n]s

[

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

time t / [s]

x(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

time t / [s]

y(t)

8


Additive Noise in the Frequency Domain

• Additive noise model in the frequency domain:

yn]

[n]

[n]

(e jΩ)

(e jΩ)

v

Ω

R

Ω

R e jΩ)(

ΩRvv

xx yy

x[

• x[n] is the signal, v[n] thecorrupting noise;

• we assume x[n] and v[n] tobe zero mean processes andmutually uncorrelated;

• measure signal quality: signal-to-noise ratio (SNR);

• signal and noise power can bedetermined from the PSDs.

• from assumptions −→ σ2yy = σ2

xx + σ2vv and Pyy(ejΩ) = Pxx(ejΩ) + Pvv(ejΩ).

9


Signal-to-Noise Ratio

• The signal to noise ratio is a power ratio:

SNR =signal power

noise power(5)

• for zero-mean signals: SNR = σ2signal/σ2

noise;

• the range of values to be measured may span several orders of magnitude (such asthe human hearing); therefore a logarithmic scale has been introduced:

SNRdB = 10 · log10

σ2signal

σ2noise

= 20 · log10

σsignal

σnoise[deciBel, dB] (6)

• Examples: σ2signal = 1, 000 · σ2

noise −→ SNRdB = 30 dB;

σ2signal = 1, 000, 000 · σ2

noise −→ SNRdB = 60 dB.

10


Analogue-to-Digital and Digital-to-Analogue Conversion

• In order to interface with the real world, we will need to convert between analogueand digital signals;

• this is accomplished by analogue-to-digital and digital-to-analogue converters(ADCs / DACs);

• analogue signals, such as a speech signal, may require a conversion into the digitaldomain for applying sophisticated source coding algorithms:

processingdigital

DACAD Canalogue

in out

analogue

• digital signals need to be converter to analogue quantities for transmission over anantenna link:

DAC AD Cdigital

in

digital

out

transmissionchannel

11


Analogue-to-Digital Conversion

• An ADC requires conditioning of the analogue signal, sampling, and quantisation:

AAF

Ts

Qx[n]x t( )

Rsignal conditioning

• sampling: the selection of the sampling period Ts will affect the possiblerepresentation of the frequency content of x(t) by x[n];

• quantisation: the word length R will influence resolution, or — as derived in slide17 — the signal-to-quantisation noise ratio (SQNR);

• signal conditioning: contains an anti-alias filter (AAF) and a gain in order to adjustthe amplitude of x(t) to the input range of the quantiser Q.

12


Sampling

• Sampling x[n] from a continuous x(t): xs(t) = x(t) ·∞∑

n=−∞δ(t− nTs);

• remembering the Fourier series theorem and the duality property of the Fouriertransform, it is clear that the frequency domain of the sampled signal xs(t) hasbeen periodised, w.r.t. ωs = 2π/Ts:

Xs(jω) =∞∑

n=−∞X(j(ω − nωs)) (7)

• hence

ωω ω ω

X(jω) (jω)sX

ω

ss s s0−2 − 2

,

13


Nyquist Rate and Aliasing

• Nyquist sampling theorem: all information of a continuous time signal is retainedif spectral repetitions do not overlap; theoretically, the analogue signal can beperfectly reconstructed from the sampled version;

ωω ω ω

X(jω) (jω)sX

ω

ss s smaxω0−2 − 2

,aliasing

ωω ω ω

X(jω) (jω)sX

ω

ss s smaxω0−2 − 2

,no aliasing

• hence the requirement:

ωs > ωN = 2 · ωmax with Nyqust rate ωN (8)

14


Signal Conditioning

• For a perfect signal representation, we have to sample a signal at least at theNyquist rate;

• in order to avoid aliasing, we must fulfil: ωmax < ωs/2;

• if the signal’s content is not exactly known, we can guarantee this by conditioningthe input signal to the ADC by an anti-alias filter:

(jω)H

ω ωω ωω

s ss− s− 0 /2/2

• this filter removes all frequency components above ωs/2, which would otherwisealias in the sampling stage.

15


Quantisation

• An R-bit quantiser approximates a straightline with 2R levels; example with R = 3and asymmetric characteristic −→

• the input signal has to use the range[−V ; +V ] as best as possible, otherwiseclippling or coarse resolution result;

• quantisation noise: maximum q/2,uniformly distributed, additive;

-V +V

q

x

binaryoutput

000

011

010

001

100

101

110

111

• quantisation noise power (no clipping!)

σ2e =

∫ q/2

−q/2

e2 · p(e)de =1q

[13e3

]q/2

−q/2

=q2

12(9)

16


Signal-to-Quantisation Noise Ratio

• Consider a uniformly distributed input signal x(t):

-V +V

( )p x

x1/2V

• therefore, the signal power is

σ2xx =

∫ V

−V

x2 · p(x)dx =1

2V

[13x3

]V

−V

=V 2

3(10)

• with (9) and q = 2V/2R, the signal-to-quantisation noise ratio (SQNR) is given by

SQNRdB = 10 · log10

V 2 · 12 · 22R−2

3 · V 2= 10 · log10 22R ≈ 6.02 ·R [dB] (11)

17


Digital-to-Analogue Conversion

• For the onversion from the digital to the analogue domain, a DAC comprises of:

R

Qx[n]

Ts

x t( )

x t( ) x[n ]signal conditioning

RCF−1

Ts

t

,

• the voltage signal at the output of Q−1 contains high frequency components, whichneed to be removed:

ωω ω ω

X(jω) (jω)sX

ω

ss s s0−2 − 2

,spectral repetitions

desired spectrum

18


Reconstruction Filter

• In order to reconstruct the analogue signal x(t), a lowpass filter is required toremove undesired high frequency components;

• this lowpass filter interpolates between adjacent sample values and is thereforetermed reconstruction filter; requirement: stopband edge at ωs/2:

(jω)H

ω ωω ωω

s ss− s− 0 /2/2

• with a perfect anti-alias filter and reconstruction filter, the concatenation of ADCand DAC imposes no loss on the signal (Nyquist);

• loss occurs due to (i) non-ideal anit-alias and reconstruction filters and (ii)quantisation.

19


Realistic ADC/DAC

• High quality analogue filters for anti-alias and reconstruction filtering require highorder and low tolerances on elements;

• Realistically, a sufficient roll-off has to be permitted, hence limiting the “ideal”reconstruction:

roll−off

| ( jω)|

ω

cut−offfrequency

gainH

• analogue components can be the most expensive part in a DSP system;

• we will study later how part of the analogue processing can be performed in thedigital domain by oversampling and Σ−∆ converters.

20

Signal Processing, 2004/05 — 3. Spectrum Estimation S. Weiss

Spectrum Estimation

• The discrete Fourier transform (DFT) has been discussed previously (see sections11 and 12 in Prof. Rogers’ notes);

• we want to consider techniques to estimate the spectra of stochastic signals;

• if x is an N -element vector taken from a tap delay line (TDL) and T an N -pointDFT matrix, then from the transform vector X

X = T · x (12)

a periodogram can be derived by squaring the Fourier coefficient; problem: if x israndom, then X will also be random;

• if we want to obtain a deterministic quantity for the spectrum of a random signal,clearly some form of expectation/averaging needs to be incorporated — either inthe time or the frequency domain.

21


Recall: Stochastic Processes

• A stochastic process is characterised by some deterministic measures:

– its probability density function (PDF) (= normalised histogram);– its moments

∫xlp(x)dx of order l;

(note that l = 1 is mean, l = 2 is variance for zero-mean process);– its autocorrelation function;

• central limit theorem: the sum of arbitrarily distributed processes converges to aGaussian PDF;

• Gaussian (or normal) PDF of a process x[n] ∈ N (µx, σ2xx):

p(x) =1√

2πσxx

e−(x−µx)2

2σ2xx (13)

22


Stationarity and Ergodicity

• Stationarity means that the statistical moments of a random process do not changeover time;

• a weaker condition is wide-sense stationarity (WSS), i.e. moments up to secondorder (mean and variance) are constant over time; this is sufficient unless higherorder statistics (HOS) algorithms are deployed;

• a stochastic process is ergodic if the expectation operation can be replaced by atemporal average,

σ2xx =

∞∫−∞

x2p(x)dx = Ex[n]x∗[n] = limN→∞

1N

N−1∑n=0

|x[n]|2 (14)

• remember: expectation is an average over an ensemble; a temporal average isperformed over a single ensemble probe!

23


Moving Average (MA) Model / Signal

• The PDF does not contain any information on how “correlated” successive samplesare;

• consider the following scenario with x[n] ∈ N (0, σ2xx) being uncorrelated (successive

samples are entirely random):

- h[n] -

x[n] y[n] = x[n] ∗ b[n]

N (0, σ2xx) N (0, σ2

yy)

• y[n] is called a moving average process (and b[n] an MA model) of order N − 1 if

y[n] =N−1∑ν=0

b[ν]x[n− ν] is a weighted average over a window of N input samples.

24


Autoregressive (AR) and ARMA Models / Signals• If the process y[n] has emerged from an uncorrelated process x[n] ∈ N (0, σ2

xx) viathe difference equation

y[n] = b[0]x[n]−N−1∑ν=1

a[ν]y[n− ν] ; (15)

it is called an N − 1 order auroregressive (“self-dependent”,[2]) signal;

• the system associated with (15), defined by its parameters b[0] and a[ν], ν =1 · · ·N − 1, is referred to as an autoregressive model.

• an MA model is often also called an all-zero model or FIR filter; an AR model is anall-pole filter; autoregressive moving average (ARMA) systems refer to a generalIIR filter;

• such a model can be very useful in characterising the stochastic process y[n] by aninnovations filter[3].

25


Filtering a Random Signal

• Consider lowpass filtering an uncorrelated Gaussian signal x[n]:

- h[n] -

x[n] y[n] = x[n] ∗ h[n]

N (0, σ2x) N (0, σ2

y)

0 50 100 150−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

time n

x[n]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

norm. angular freq. Ω/π

|H(e

jΩ)|

0 50 100 150−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time n

y[n]

• the output will have Gaussian distribution, but the signal only changes smoothly:neighbouring samples are “correlated”. We need a measure!

26


Auto-Correlation Function I

• The correlation between a sample x[n] and a neighbouring value x[n− τ ] is givenby

rxx[τ ] = Ex[n] · x∗[n− τ ] = limN→∞

1N

N−1∑n=0

x[n] · x∗[n− τ ] (16)

• For two specific specific lags τ = 3 (left) and τ = 50 (right), consider:

0 50 100 150−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time n

x[n]

, x[n

+3]

0 50 100 150−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time n

x[n]

, x[n

+50

]• the curves on the left look “similar”, the ones on the right “dissimilar”.

27


Auto-Correlation Function II

• For lag zero, note:

rxx[0] = limN→∞

1N

N−1∑n=0

x[n] · x∗[n] = σ2x + µ2

x (17)

• This value for τ = 0 is the maximum of the auto-correlation function rxx[τ ];xxr [τ]

τ

• large values in the ACF indicate strong correlation, small values weak correlation;

28


Auto-Correlation Function III

• If a signal has no self-similarity, i.e. it is “completely random”, the ACF takes thefollowing form:

xxr [τ]

τ

• If we take the Fourier transform of rxx[τ ], we obtain a flat spectrum (or a lowpassspectrum for the ACF on slide 28);

• due to the presence of all frequency components in a flat spectrum, a completelyrandom signal is often referred to as “white noise”.

29


Power Spectral Density

• The power spectral density (PSD), Rxx(ejΩ), defines the spectrum of a randomsignal:

Rxx(ejΩ) =∞∑

τ=−∞rxx[τ ] ejΩτ (18)

• PSD and ACF form a Fourier pair, rxx[τ ] —• Rxx(ejΩ), therefore

rxx[τ ] =12π

∫ π

−π

Rxx(ejΩ) e−jΩτ dΩ (19)

• note that the power of x[n] is (similar to Parseval)

rxx[0] =12π

∫ π

−π

Rxx(ejΩ) dΩ (= scaled area under PSD) (20)

30


PSD – Examples

• PSD for uncorrelated (“white”) zero mean noise:

σxxr [τ]

x2

τ

—• σe jΩ)

Ω0 π−π

x2

( xxR

• PSD for correlated zero mean noise:

σ xxr [τ]x2

τ—•

e jΩ)2π σx

2area:

0

Ωπ−π

( xxR

31


Cross-Correlation

• The cross-correlation function between two signal x[n] and y[n] is defined analogousto (16):

rxy[τ ] = Ex[n] · y∗[n− τ ] = limN→∞

1N

N−1∑n=0

x[n] · y∗[n− τ ] (21)

• note: ryx[τ ] = r∗xy[−τ ] and especially rxx[τ ] = rxx[−τ ]

i.e. the auto-correlation function is symmetric, while the cross-correlation functionis not;

• for uncorrelated signals:

rxy[τ ] = Ex[n] · y∗[n− τ ] = Ex[n] · Ey∗[n− τ ] = µx µ∗y (22)

32


Examples for Applying the Cross-Correlation

• Delay estimation. Imagine we send a random pulse x[n] and wait for the reflectedreturn signal y[n]:

x[n] n

y[n]n

T T

τ

r [τ]yx

• “Matched filtering”. Compare a received signal with an expected waveform; thecross-correlation will be maximum if the received signal matches up with a desiredsequence.

33


PSD and Innovations Filter

• Consider again filtering a random signal x[n] with a filter having an impulseresponse h[n] (either MA, AR, or ARMA):

- h[n] -

x[n] y[n] = x[n] ∗ h[n]

rxx[τ ] ryy[τ ]

Rxx(ejΩ) Ryy(ejΩ)

• relation between x[n] and y[n] is given by convolution: y[n] =∞∑

ν=−∞h[ν] x[n− ν];

• we are looking for the relations between rxx[τ ] and ryy[τ ] and between Rxx(ejΩ)and Ryy(ejΩ).

34


• The cross-correlation is:

ryx[τ ] = limN→∞

1N

N−1∑n=0

y[n] · x∗[n− τ ] (23)

= limN→∞

1N

N−1∑n=0

·

( ∞∑ν=−∞

h[ν] x[n− ν]

)x∗[n− τ ] (24)

=∞∑

ν=−∞h[ν] lim

N→∞

1N

N−1∑n=0

x[n− ν] x∗[n− τ ] (25)

=∞∑

ν=−∞h[ν] lim

N→∞

1N

N−1∑n=0

x[n] x∗[n− τ + ν] (26)

=∞∑

ν=−∞h[ν] rxx[τ − ν] = h[τ ] ∗ rxx[τ ] (27)

• note: rxy[τ ] = r∗yx[−τ ] = h∗[−τ ] ∗ rxx[τ ]

35


• Going further:

ryy[τ ] = limN→∞

1N

N−1∑n=0

y[n] · y∗[n− τ ] (28)

= limN→∞

1N

N−1∑n=0

y[n]∞∑

ν=−∞h∗[ν] x∗[n− ν] (29)

=∞∑

ν=−∞h∗[ν] lim

N→∞

1N

N−1∑n=0

y[n] x∗[ν − n + τ ] (30)

=∞∑

ν=−∞h∗[ν] ryx[ν + τ ] = h∗[−τ ] ∗ ryx[τ ] (31)

= h∗[−τ ] ∗ h[τ ] ∗ rxx[τ ] (32)

36


PSD and Innovations Filter II

• Hence, if a stochastic process x[n] is filtered:

- h[n] -

x[n] y[n] = x[n] ∗ h[n]

rxx[τ ] ryy[τ ]

Rxx(ejΩ) Ryy(ejΩ)

• we have

ryy[τ ] = h∗[−τ ] ∗ h[τ ] ∗ rxx[τ ] (33)

Ryy[τ ] = H∗(ejΩ) H(ejΩ) Rxx[τ ] (34)

= |H(ejΩ)|2 Rxx[τ ] (35)

• note for innovations filter (uncorrelated input x[n] ∈ N (0, 1)): Ryy[τ ] = |H(ejΩ)|2.

37


Averaged Periodogram I

• The PSD is a deterministic quantity of a random process, while the periodogram israndom;

• averaging of the periodogram is required to eliminate (or at least reducerandomness);

• method 1: take N -point DFT of x[n], and average over a neighbourhood of squaredFourier coefficients;

π

jΩ)|2 |X(e jΩ)|2,

Ω

|X(e

• method 2: calculate a number of periodograms and average across the the frequencybins (Fourier coefficients).

38


Averaged Periodogramme II

2

[n]

| . |2

DFT

| . |2

DFT

| . |2

|X(e jΩ)|2

n

DFT

averaging

jΩ)|2 jΩ)|2 jΩ)|2|X (e |X (e|X (e0 1

x

39


Recall Spectral Leakage

• The DFT “periodises” the data, which is likely to create aberrations;

• N = 32, by chance the window ends fit:

0 20 40 60 80−1

0

1

x[n]

0 5 10 150

10

20

|X[k

]|

n k

N

• N = 27, discontinuties arise at the window edges, causing the main peak to “leak”:

0 20 40 60 80−1

0

1

x[n]

0 5 100

5

10

|X[k

]|N

n k

40


Recall Windowing

• DFT applied to a segment of rectangularly windowed sinusoidal data:

10 20 30 40−20

0

20

40

timen

k

leakage

mag

nitu

de /

[dB

]

• DFT after application of a Hamming window to the same data:

10 20 30 40−20

0

20

40

timen

kmag

nitu

de /

[dB

]

• with windowing, the spectral leakage is reduced at the cost of a widened main lobe.

41


Practical PSD Calculation

DFT

[n]

|X(e jΩ)|2

| . |2 | . |2 | . |2 | . |2 | . |2 | . |2

n

averaging

DFT DFT DFT DFT DFT

x

42


Relation between Averaged Periodogram and PSD

• Note the differences and correspondences between the PSP and the averagedperiodogram:

E·x[n] −→ rxx[τ ]

DFT ↓ ↓ DFT

|X(ejΩ)|2 −→ Rxx(ejΩ)E·

(36)

• spectral estimation (approximations of the PSD) are usually based on the averagedperiodogram method outlined in slide 42.

43


Summary• We have defined PSD theoretically and the averaged periodogram as a

computational technique to estimate it;

• usually the longer the data sequence x[n] available, the more confident (accurate)this estimate is;

• we will define a parametric method for spectrum estimation, trying to guess at theinnovations filter behind the signal at hand;

• for background and further reading:

[1] B. Girod, R. Rabenstein, and A. Stenger. Signals and Systems. J. Wiley & Sons, 2001.

[Good explanation of auto- and cross-correlation, and the PSD]

[2] T. K. Moon and W. C. Stirling. Mathematical Methods and Algorithms for Signal Processing.

Prentice Hall, 1999.

[3] A. Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill, 3rd ed.,

1991.

[A classic text.]

44

Signal Processing, 2004/05 — 4. Digital Filtering S. Weiss

Digital FIR Filtering

• Finite impulse response (or moving average, MA) filter:

xx [ ]n x [ −1] x [ ] x [ − +1]

[ −1]

n n−2 n N

h h h h N[ ]0 [ ]1 [ ]2

xx [ ]n

• tap delay line (TDL) of length N ;

• described by difference equation, y[n] =∑N−1

ν=0 bnx[n− ν]

• no feedback: inherently stable.

45


Impulse Response

• For an FIR filter, the impulse response h[n] (earlier stated as zero-state response,i.e. the TDL is initialised to zero) is given by its filter coefficients:

n

n[ ]h

[n]y

n

N0

x

0

[n]

• the impulse response has finite support with h[n] = 0 ∀n ≥ N ;

• the impulse response can give direct clues about multipath propagation, or echosand reverberation.

46


Frequency Response

• Frequency response H(ejΩ): measures the gain of the system in the steady state:

• impulse and frequency response are a Fourier pair: h[n] —• H(ejΩ).

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

time index n

h[n]

—•

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

normalised angular freq. Ω / 2π

|H(e

jΩ)|

47


Magnitude and Phase Response

• The magnitude decides whether the system has e.g. lowpass, highpass, or bandpasscharacteristic;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5


|H(e

jΩ)|

• phase contains a number of exciting characteristics of a filter:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−40

−20

0


∠ H

(ejΩ

) /r

ad

48


Group Delay

• from H(ejΩ) = |H(ejΩ)|ejΦ(Ω), note for the group delay g(Ω):

g(ejΩ) = − d

dΩΦ(Ω) (37)

0 0.2 0.4 0.6 0.8 1−80

−60

−40

−20

0


∠ H

(ejΩ

) /r

ad

−→

0 0.2 0.4 0.6 0.8 110

11

12

13

14


g(ejΩ

) / s

ampl

es

• the group delay tells us by how long certain frequency components will be delayedwhen propagating through the filter;

• note the difference between linear phase and non-linear phase systems.

49


z-Transform

• Reconsider FIR filter:

• z-transform of difference equation:

Y (z) = b0X(z) + b1z−1X(z) + b2z

−2X(z) + · · ·+ bN−1z−N+1X(z) (38)

H(z) =Y (z)X(z)

=N−1∑i=0

biz−i (39)

• the z-transform with z = ejΩ+α can capture the transient behaviour of a system;

• note that H(ejΩ) = H(z)|z=ejΩ.

50


z-Transform — Some Useful Properties

• Assume a z-transform pair h[n] •— H(z) =∑∞

i=−∞ hi z−i

• time shift:h[n−∆] •— H(z) z−∆ (40)

• complex conjugation:

h∗[n] •— H∗(ejΩ) =∞∑

i=−∞h∗i z−i (41)

• time reverse:

h[−n] •— H(z−1) =∞∑

i=−∞hi zi (42)

51


Transfer Function

• We can factorise the transfer function H(z):

H(z) = b0 + b1z−1 + · · · bN−1z

−N+1 (43)

= b0(1− β0z−1)(1− β1z

−1) · · · (1− βN−2z−N+2) (44)

• the roots βi of H(z) are called the zeros of the transfer function;

• often a pole-zero plot is used for visulisation:

52


Real Valued Systems

• If the impulse response only contains real-valued coefficients, the zeros βi must beeither real valued, or occur as complex conjugate pairs (as seen in slide52;

• we could factorise the system into a sequence of first order sections:

X(z) - b0- 1− β0z

−1 · · · - 1− βN−2z−1 - Y (z)

• a complex multiplication take 4 real valued multiplications; therefore, forimplementational purposes, we would always prefer (44) over (44).

53


Minimum and Maximum Phase

• For a minimum phase system, all zeros lie inside the unit circle;

• a maximum phase system has all zeros outside the unit circle;

• interesting: if h[n] is minimum phase, then h[−n] is maximum phase;

• example for H1(z) = 1 + 12z−1 and H2(z) = 1

2 + z−1:

H1(z)H2(z)H3(z) = H1(z)H2(z)

•—

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

normalised angular freq. Ω / 2π∠

Hi(e

jΩ)

/rad

54


Linear Phase Filters

• recall: linear phase means constant group delay;

• a linear phase system must have a z-transform of the form (including a possibledelay for causality)

H(z) = Hmin. phase(z) Hmin. phase(z−1) Πm(1− ejφmz−1) (45)

• any zero of a linear phase system must have a reciprocially matching partner (unlesson the unit circle);

• linear phase iff impulse response is symmetric: recall that hmin. phase[n] ∗hmin. phase[−n] will give an auto-correlation type result.

55


Summary

• Note that certain properties of a system are best assessed in a specific domain:either impulse response, frequency domain, or transfer function;

• phase information is often vital, e.g. if we need to invert the system (for equalisationor control purposes);

• most FIR filter designs yield symmetric impulse response and hence are linear phase;

• we will consider such filter designs next ...

56

Signal Processing, 2004/05 — 5. FIR and IIR Design S. Weiss

FIR and IIR Design

• Design characteristics: passband width, transition band width, stopband edge,stopband attenuation, and passband ripple:

0 dB

-3 dB

cΩ

e )Ωj(H

transition

ripple

stopband

Ωπ0

band width

attenuation

passband

frequencycutoff

passband

width

stopbandedge

• generally: high quality requires high number of coefficients.

57


Different Frequency Responses

• Four different basic filter types as defined by their magnitude response: (a) lowpass,(b) highpass, (c) bandpass, and (d) bandstop filter:

e )Ωj(H

Ω

e )Ωj(H

Ω

e )Ωj(H

Ω

e )Ωj(H

Ωπ π

π π

0 0

00

(a)

(c) (d)

(b)

58


FIR Filter Design: IDFT and Windowing

• We often have an idea what the magnitude response |H(ejΩ)| of the desired filtershould look like; additionally, we need to “invent” a phase response (e.g. linear);

• the frequency response is inversely Fourier transformed;

H(e jΩ) h[n]

Ωn

• the resulting time domain response is an approximation of the desired impulseresponse (holding the filter coefficients);

• some windowing may be required in order to enforce finite support in the timedomain.

• a variety of design approaches based on minimax or least squares methods exists.

59


FIR Design Examples

• Comparison of two filter designs with 16 and 48 coefficients, respectively:

0 5 10 15 20

0

0.1

0.2

h 1[n]

0 10 20 30 40 50

0

0.1

0.2

h 2[n]

time index n

0 0.1 0.2 0.3 0.4 0.5−80

−60

−40

−20

0

Ω / 2π

|H1(ejΩ)|

|H2(ejΩ)|

• both designs have identical passband edge at Ω = 0.1π.

60


IIR Filtering

• Various ways to represent the difference equation as a flow graph:

y[n] =L−1∑l=0

al x[n− l] +J−1∑j=1

bj y[n− j] (46)

[ ]1b [ ]2b[ ]0b

xx [ ] x [ -1 ] x [ ]-2 x [ - +1]

b[ -1 ]

n n n n L

L

a[J-1 ] a[ ]2 a[ ]1

y [ ]

y [ - J+1] y [ ]-2 y [ -1 ]n n n

n

adder

multiplier

delay element

61


IIR Stability

• transfer function:

H(z) =Y (z)X(z)

=b0 + b1z

−1 + · · · bL−1z−L+1

1− a1z−1 − a2z−2 − · · · − aJ−1z−J+1(47)

= b0(1− β0z

−1)(1− β1z−1) · · · (1− βL−2z

−1)(1− α0z−1)(1− α1z−1) · · · (1− αJ−2z−1)

(48)

• : note that H(z) can be spit into a cascade of FIR and recursive first-order sections;

• each recursive section 11−αz−1 must be stable; note geometric series

11− αz−1

=∞∑

i=0

αi z−i for |a| < 1 (49)

• poles must be inside the unit circle!

62


IIR Filter Design

• IIR filter design is based on analogue techniques (Butterworth, Chebychev etc);

• the analogue design is transformed from the Laplace- to the z-domain by means ofthe bilinear transform:

s =1Ts

1− z−1

1 + z−1(50)

• IIR filters generally achieve high quality filters with less coefficients than FIR.

• Appreciate the response length (again geometric series):

11− az−1

=∞∑

i=0

ai z−i for |a| < 1 (51)

63


FIR/IIR Comparison

• Comparison between FIR and IIR filters (both 24 coefficients):

0 0.1 0.2 0.3 0.4 0.5−80

−60

−40

−20

0

20FIR

|H(e

jΩ)|

/ [d

B]

0 0.1 0.2 0.3 0.4 0.5−80

−60

−40

−20

0

20FIR quantised, B=8 bit

norm. ang. freq. Ω/2π

|H(e

jΩ)|

/ [d

B]

0 0.1 0.2 0.3 0.4 0.5−80

−60

−40

−20

0

20IIR

0 0.1 0.2 0.3 0.4 0.5−80

−60

−40

−20

0

20IIR quantised, B=8 bit


0 0.1 0.2 0.3 0.4 0.511.5

11.5

11.5

11.5

11.5

11.5

11.5FIR

g(ejΩ

) / [

sam

ples

]

0 0.1 0.2 0.3 0.4 0.511.5

11.5

11.5

11.5

11.5FIR quantised, B=8 bit

g(ejΩ

) / [

sam

ples

]


0 0.1 0.2 0.3 0.4 0.5−20

0

20

40

60

80

100IIR

0 0.1 0.2 0.3 0.4 0.50

50

100

150IIR quantised, B=8 bit


• note that quantisation can affect an IIR filter badly (instability)!

64


Computational Complexity I

• Recall for general IIR filter:

y[n] =L−1∑l=0

blx[n− l] +J−1∑j=1

ajy[n− j] = bHxn + aHyn−1 (52)

• whereby vector notation:

xn = [x[n] x[n− 1] · · ·x[n− L + 1]]T (53)

yn−1 = [y[n− 1] y[n− 2] · · · y[n− J + 1]]T (54)

b = [b[0] b[1] · · · b[L− 1]]H (55)

a = [a[1] a[2] · · · a[J − 1]]H (56)

(57)

65


Computational Complexity II

• Hence required steps per sampling period:

– update TDL (memory moves– calculate L + J multiply-accumulates (MAC)

• Example 1. Sampling rate fs = 8 kHz; we want to model the impulse responseof an acoustic system with a T = .5 s duration, and therefore need T · fs = 4000coefficients. Total complexity: C = 4000fs = 32 MMAC/s;

• Example 2. We double the sampling rate to wideband audio with fs = 16 kHz;we require 8000 coefficients and yield a total complexity of C = 128 MMAC/s;

• Note: doubling the sampling rate means quadrupling the complexity!

• State-of-the-art digital signal processors perform approximately 1 GMAC/s.

66


Summary on FIR/IIR Filtering

• Characterisation by impulse response, frequency response, or transfer function;

• Be aware of properties such as causality and minimum/maximum/non-minimumphase;

• Stability. FIR is strictly stable, IIR is not;

• Quality/Complexity. IIR filters need less coefficients than FIR.

• Linear Phase. For FIR filters a symmetric impulse response is necessary andsufficient for linear phase; IIR are never linear phase (although some yield anapproximation in the passband);

• Implementation. Both FIR and IIR are sensitive to fixed point implementation,but IIR has larger dynamics in its filter coefficients and may even become unstablewhen quantised.

67

Signal Processing, 2004/05 — 6. Adaptive Filtering S. Weiss

Adaptive Filtering

• Previously we have designed filters with specific characteristics; reconsider an FIRfilter with L coefficients w[n]:

ny[ ]

nd[ ]

e n[ ]

nx[ ]

L-10 1 2w w w w

• here the adjustment of the coefficients w[n] may be dependent on the environment,and the coefficients may be variable over time.

68


Generic Adaptive Filtering

• The architecture of a generic adaptive filter comprises of an input signal x[n], adesired signal d[n], a filter output y[n], an error signal e[n], and the filter impulsresponse w[n];

ny[ ]

nd[ ]

e n[ ]

nx[ ]

L-10 1 2w w w w

• Aim of an adaptive filter: the filter w[n] forms an output y[n] from an inputx[n], such that, when subtracted from the desired signal d[n], the reslting error isminimised in a suitable sense.

69


Adaptive Filter Architectures

• A generic adaptive filter can be applied in different architectures:

• system identification

ny[ ] e n[ ]

nd[ ]

nx[ ]w

c

• inverse system identification /equalisation

ny[ ] e n[ ]nx[ ]c

nd[ ]

w

• prediction

ny[ ] e n[ ]nx[ ]

nd[ ]

w

• noise cancellation

ny[ ] e n[ ]nx[ ]

nd[ ]

w

n[ ]v + ]nx[

70


System Identification

• We want to identify an unknown LTI system with impulse response c[n]:

ny[ ] e n[ ]

nd[ ]

nx[ ]w

c

• if the adaptive filter converges and e[n] is (ideally) zero, then the input-outputbehaviour of c[n] and w[n] is identical;

• if x[n] is sufficiently broadband, we have identified c[n] by means of a suitablemodel contained in w[n].

71


Inverse System Identification / Equalisation

• We place the unknown system and the adaptive filter in series:

ny[ ] e n[ ]nx[ ]c

nd[ ]

w

• if the error signal is minised, then w[n] represents the inverse system of c[n];

• if c[n] is non-minimum phase, we need to permit a delay ∆, such that c[n]∗w[n] ≈δ[n−∆].

72


Prediction

• Due to delaying the input signal x[n], the desired signal d[n] is a “future version”of the filter input:

ny[ ] e n[ ]nx[ ]

nd[ ]

w

• the adaptive filter should only be able to predict “periodic” or correlated componentswith d[n];

• the error signal e[n] would in this case contain any remaining uncorrelated signalcomponents.

73


Adaptive Noise Cancellation

• consider a noise corrupted signal d[n] (e.g. helicopter pilot’s microphone signal):

ny[ ] e n[ ]nx[ ]

nd[ ]

w

n[ ]v + ]nx[

• if we can obtain a reference signa x[n] of the noise, the adaptive filter should beable to cancel the noise component from e[n], provided that the noise in x[n] andd[n] is linearly related.

74


Acoustic Echo Cancellation

• hands-free telephony — full-duplex mode suffers from acoustic feedback:

[n]y

[n]e

[n]d

[n]x

room

echo

near endspeaker

speech+ echo

far endspeaker

roommodel replica

echo

adaptive digital filter – acoustic feedback is

very disturbing forthe far end speaker

– by identifying theroom acoustics (overthe frequency rangeof speech), theadaptive filter canproduce a replica ofthe echo;

• due to the reverberation times of a room, which the adaptive filter has to match,this is a very computationally demanding task.

75


How Can We Optimise the Filter Coefficients?

• In order to calulate an optimal filter wopt[n], the mean square error (MSE) Ee2[n]

is a good criterion to be minimised;

• Formulation of error signal e[n]:

e[n] = d[n]− y[n] = d[n]−L−1∑i=0

wi · x[n− i] = d[n]−wH · xn

• with definitions:w =

[w0 w1 · · · wL−1

]T;

xn =[x[n] x[n− 1] · · · x[n− L + 1]

]T;

• (we could do the derivation for complex valued signals, but for simplicity here stickwith the real valued case.)

76


Optimal Filter — MSE Formulation

• Mean Square Error formulation (note that the optimum filter wopt is deterministic!):

Ee2[n]

= E

(d[n]−wT · xn)2

= E

d2[n]

− 2wT · Ed[n] · xn+ wT · E

xn · xT

n

·w

= σ2d − 2wT · p + wT ·R ·w

with covariance matrix R = Exn · xT

n

and correlation vector p = Ed[n] · xn;

• note: wT · xn is a scalar, hence wT · xn = (wT · xn)T = xTn ·w;

• important: the expression for Ee2[n]

is quadratic in the filter coefficients w;

hence optimisation should lead to a unique extremum!

77


Covariance Matrix & Cross-Correlation Vector

• Given the auto-correlation sequence rxx[τ ] and the cross-correlation sequencerxd[τ ]:

rxx[τ ] =1N

N−1∑n=0

x[n] · x[n− τ ] rxd[τ ] =1N

N−1∑n=0

d[n] · x[n− τ ]

• the covariance matrix R and cross-correlation vector p can be written

R =

rxx[0] rxx[1] · rxx[L−1]rxx[1] rxx[0] · rxx[L−2]

... . . . ...rxx[L−1] rxx[L−2] · · · rxx[0]

p =

rxd[0]rxd[1]

...rxd[L−1]

• the Wiener-Hopf solution requires estimation of R and p, and a matrix inversion

(numerically costly and potentially unstable!).

78


MSE Surface

• Quadratic in the filter coefficients w, the surface of the MSE ξ = Ee2[n]

is a

hyperparabola in L + 1 dimensional hyperspace;

• example for L = 2 coefficients:

w0,opt

w1,optw0

w1

ξ

79


Optimum Filter — Minimum Mean Squared Error

• Standard optimisation procedure to achieve minimum MSE:

∂

∂wEe2[n]

!= 0

• Inserting the previous expression for the MSE:

∂

∂wEe2[n]

= 0− 2p + 2R ·w

• if R is invertible, then the optimum filter coefficients wopt are given by (Wiener-Hopfequation):

wopt = R−1 · p

80


Optimum Filter — Minimum Mean Squared Error

• Standard optimisation procedure to achieve minimum MSE:

∂

∂wEe2[n]

!= 0

• Inserting the previous expression for the MSE:

∂

∂wEe2[n]

= 0− 2p + 2R ·w

• if R is invertible, then the optimum filter coefficients wopt are given by (Wiener-Hopfequation):

wopt = R−1 · p

81


Viability of Wiener-Hopf

• Confident estimation of R and p requires long time windows;

• inversion of R is of order O(L3) — very complex!

• Wiener-Hopf requires R to be invertible;

• even if R has full rank, it may be ill-conditioned.

82


Iterative Solutions

• In order to bypass the problems of Wiener Hopf, we will utilise iterative approachesto converge towards the optimal solution;

– Starting from an initial guess w[0],gradient descent techniques followthe negative gradient of the MSEcost function:

w[n + 1] = w[n]− µ∇ξ[n]

– µ is called the “step size” of thealgorithm and controls convergence. . .

w0,opt

w1,optw0

w1

ξ

large µ

small µ

83


Gradient Descent Techniques

• We require the gradient ∇ξ[n] = −2p + 2Rw[n] — see slide 81

• therefore:w[n + 1] = w[n] + 2µ(p−Rw[n]) (58)

• this is also know as the method of steepest descent;

• note I: matrix inversion is no longer required;

• note II: we still need to estimate R and p.

84


Stochastic Gradient Techniques

• We omit expectations and minimise the instantaneous squared error:

we2[n] = −2e[n] · xn (59)

• this results in the least mean squares (LMS) algorithm:

w[n + 1] = w[n] + 2µe[n] · xn (60)

• the gradient estimate is noise, but through itertion, on average the gradient estimatepoints into the right direction.

85


Least Mean Squares Algorithm

ny[ ]

nd[ ]

e n[ ]

nx[ ]

L-10 1 2w w w w

• Steps for the LMS adaptive filter:(1) update tap delay line vector x[n](2) calculate filter output: y[n] = wH[n]x[n](3) calculate error: e[n] = d[n]− y[n](4) LMS update: w[n + 1] = w[n] + 2µe[n] · xn

• Complexity: 2L multiply accumulate (L for scalar product, L for LMS update) persampling period.

86


LMS Convergence

• The step size µ controls the convergence of the LMS;

• intuitive from MSE surface plot: small (large) µ −→ slow (fast) convergence withhigh (low) final accuracy;

• Without proof: convergence can be guaranteed if

0 < µ <1

Lσ2xx

(61)

where L is the filter length and σ2xx the variance of the input signal.

87


LMS System Identification Example I

• Matlab code for LMS algorithm in system identification:

x = randn(1,1000); % input signalc = [1 0 .5 -.3]; % ‘‘unknown’’ systemd = filter(c,1,x); % desired signalL = 10; % filter lengthmu = 0.01; % step size

w = zeros(L,1); % weight vector initialisationfor n = L:1000,

Xtdl = x(n:-1:n-L+1)’; % update TDLy(n) = w’*Xtdl; % calculate filter outpute(n) = d(n)-y(n); % calculate errorw = w + 2*mu*e(n)*Xtdl; % LMS weight update

end;

88


LMS System Identification Example II

• Convergence of error:

0 100 200 300 400 500 600 700 800 900 1000

−300

−200

−100

0

iterations n

|e[n

]| / d

B

• The error reaches Matlab’s floating point accuracy of approximately 10−16 — thisis numerically zero.

• after convergence, the filter coefficients match the impulse response of the unknownsystem c[n].

89


Channel Equalisation Example I

• Assume a minimum phase channel C(z) = 1 + 0.5z−1;

• the inverse of this channel is given by

W (z) =1

C(z)=

11 + 0.5z−1

(62)

• we can use the geometric series, 11−a =

∑∞n=0 an for |a| < 1, to find an FIR

approximation of the inverse:

W (z) =∞∑

n=0

(−.5)nz−n = 1− .5z−1 + .25z−2 − 0.125z−3 + · · · (63)

• For uncorrelated input, an adaptive filter of length L would identify the first Lelements of this geometric series.

90


Channel Equalisation Example II

• Now assume a maximum phase channel C(z) = .5 + z−1;

• the inverse of this channel is unstable

W (z) =1

C(z)=

21 + 2z−1

= 2− 4z−1 + 8z−2 − · · · (64)

and the “geometric series” diverges; alternatively

W (z) =z

1 + 0.5z= z

∞∑n=0

(−.5)nzn = z − .5z1 + .25z2 − · · · (65)

converges but is anti-causal.

91


Channel Equalisation Example II ctd.

• If we permit a delay of e.g. 3 samples, z−3, such that

W (z) =z−2

1 + 0.5z= z−2

∞∑n=0

(−.5)nz (66)

= z−2 − 0.5z−1 + 0.25− 0.125z1 + · · · (67)

an adaptive filter can identify the 3 causal coefficients of the geometric series.

• hence, for equalisation or inverse system identification, in general we have to permita delay for the system C(z)W (z);

• rule of thumb: not knowing anything about thye phase properties of C(z),practisioners choose a delay of L/2.

92


Channel Equalisation Example III

• Now consider a channel C(z) = 1− z−1 — this is a highpass filter with a zero atz = 1 cancelling the dc (f = 0) frequency component;

• the inverse system would attempted to provide a very high (ideally infinite) gain atfs/2 — this is very dangerous!

• inspect the covariance matrix R; for uncorrelated input u[n] to the channel, theautocorrelation function of x[n] is given by

rxx[τ ] = c[τ ] ∗ c[−τ ] (68)

and is likely to be ill-conditioned.

• try to establish what filter response the LMS algorithm converges to.

93


Channel Equalisation Example III ctd.

• In a realistic scenario, a channel suffers from additive white Gaussia noise v[n],such that

x[n] = c[n] ∗ u[n] + v[n] (69)

whereby v[n] is independent of u[n];

• therefore for the autocorrelation function due to independence:

rxx[τ ] = c[τ ] ∗ c[−τ ] + rvv[τ ] (70)

• for the covariance matrix R, we obtain

Rxx = Rcc + Rvv (71)

whereby Rcc is the covariance matrix of the noiseless channel output, and Rvv =σ2

vvI the covariance matrix of the AWGN.

94


Channel Equalisation Example III ctd.

• Comparing Rcc and Rxx, the conditioning of the matrix is improved (

Rxx = QΛQH︸︷︷︸Rcc

+σ2vvI = Q

λc,0 + σ2

vv 0 00 λc,1 + σ2

vv 0 0... . . . ...0 0 λc,L−1 + σ2

vv

QH

(72)

• simulate the previous example with channel noise: in the presence of channelnoise, the adaptive filter strikes a balance between inversion of C(z) and noiseamplification;

• the adaptive filter provides a minimum mean square error (MMSE) solution ratherthan an inversion: the system inverse is “regularised” by the AWGN.

95


Summary / Further Study

• Adaptive filters tuneable filter coefficients and can be applied in situation whereexactly desired response (highpass, lowpass, etc) is not known a priori;

• many other popular adaptive algorithms exist beyond the LMS;

• for further reading:

[1] B. Widrow and S.D. Stearns Adaptive Signal Processing. Prentice Hall, 1985.[Classical and easy to read text.]

[2] S. Haykin Adaptive Filter Theory. Prentice Hall, 1987, 1991, and 1996.[A good mathematical treatment — strangely the 2nd edition is consideredbetter than the third.]

96

Signal Processing, 2004/05 — 7. Multirate Signal Porcessing S. Weiss

7. Multirate Signal Processing

• In a multirate system, various components run at different sampling rates;

• Audio DSP: you may need to convert from professional audio (96kHz) down to HIFI(e.g. digital audio tape, 48kHz) or “wideband” audio (teleconferencing, 16kHz);

• Software defined radio (SDR): you may need to create a digital IF signal (sampledat, say, 75MHz) from a UMTS baseband signal (5MHz sampling rate);

Similarly, in an SDR receiver, you may have to extract a symbol stream at 5MHzfrom a 75MHz IF signal.

97


Multirate Operations

• A multirate system usually comprises of digital filters, decimators, and expanders;

• anti-alias filtering followed by N -fold decimation (downsampling) to lower thesampling rate by a factor N :

- anti-alias filter -

&%'$↓ N -

• N -fold expansion or up-sampling followed by interpolation filtering to increase thesampling rate by a factor of N :

-

&%'$↑ N - interpolation filter -

98


Decimation — Time Domain Desciption• To lower the sampling rate by a factor of N , a decimator or downsampler only

retains every N th sample — other samples are discarded; exampled for N = 4:

0 5 10 15 20 25 30 35 40−0.5

0

0.5

time index n

x[n]

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

time index m=n/4

x d[m]

• if the signal is suitably bandlimited, we loose no information in the original signal;also, the exact sampling point can be chosen arbitrarily.

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Decimation — Frequency Domain Description

• similar to sampling on slide 13, resampling “periodises” the spectrum with respectto the new sampling rate ω′s = ωs/N ; example for N = 4 (with unnormalisedangular frequencies):

X(jω)

ωω(jω)sX

ω ωωωω

ω

s s ss

s s

−4− 0 4

0−

• note: if the original signal is not suitably bandlimited, aliasing may occur.

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Anti-Alias Filtering

• A digital anti-alias filter can be employed to filter the signal prior to decimation:

H ejΩ( )

πΩ

ππN Nπ− // 0−

• it is difficult to create a brick-wall type filter; however, in the digital domain we canachieve a high quality filter design at a lower “cost” than in the analogue domain.

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Expansion — Time Domain Description• To increase the sampling rate by a factor of N , an expander or upsampler fill N −1

zero sample in between every other sample; exampled for N = 4:

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

time index m

x[m

]

0 5 10 15 20 25 30 35 40−0.5

0

0.5

time index n

x e[n]

• note that our signal is now impulsive . . . our signal can be expected to containsome high frequency components.

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Expansion — Frequency Domain Description

• Expansion simply re-scales the frequency axis:

ωωω

ωωω

jω)X (e

X(jω)

ωω

−

−4

s s

s s− 4 ss 0

0

• in the expanded signal, we see N − 1 spectral repetions in addition to the original(baseband) signal;

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Expansion — Interpolation Filter

• A digital interpolation or reconstruction filter can be employed to retireve thebaseband portion of the signal after expansion:

H ejΩ( )

πΩ

ππN Nπ− // 0−

• this is identical to the conditions required of the anti-alias filter.

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Oversampled ADC

• Consider sampling an analogue signal at fs = NfN rather than fs = fN :

]AAF

Ts

x t( )AAF N

x[n]

Rsignal conditioning

N

Qx[m

• the analogue anti-alias filter can now be relaxed:H(jω)

πN fN

ωπfN0

• the transition band of the anti-alias filter can be much wider than in the non-oversampled case; to compensate, we will postprocess the oversampled digital x[m]by (much cheaper) digital operations.

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Digital Postprocessing in Oversampled ADCs

• Requirement for the digital anti-alias filter is a tight cut-off at Ωc = π/N :

Ωπ

noise

signal

πN Nπ

H ej( Ω)

π 0− //−

• this is easier to achieve than a high quality analogue filter;

• additionally, the digital AAF reduces the quantisation noise power (q2/12) by afactor 1/N (recall that the area under PSD is a measure for the power!).

106


Virtual Increase in Word Length

• Assume that the original signal was quantised with a word length R;

• the quantisation noise power after the digital anti-alias filter has removed theout-of-band noise for N = 4:

σ2q =

14

q2

12=

(q/2)2

12(73)

• if the quantisation step size is halved, we have gained an extra bit in resolution;

• for every quadrupling of the oversampling ratio N , we achieve an extra bit inresolution.

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Trade-off between Sampling and Wordlength

• We can oversample our analogue signal by 4∆R to achieve ∆R extra bits inresolution after digital postprocessing;

• to achieve 16 bit quality from a single bit quantiser, we would have to oversampledby a factor 415;

• single bit converters exist, but use an additional technique, whereby the quantisationnoise is “pushed” towards higher frequencies and can be better suppressed by adigital lowpass anti-alias filter.

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Sigma-Delta Converter

• A sigma-delta converter is a mixed analogue-digital device, which provides a low-bit(often 1 bit) output:

+ 1 bit outputinput

DAC

integrator comparator

−

• The input sigma is passed through an integrator (lowpass filter), and the output isconverted with single bit accuracy, i.e. only retaining the sign;

• the output bit-stream is converted into a voltage signal by a DAC and subtractedfrom the input.

•

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Discrete Time Model

• In order to analyse the sigma-delta circuit, we approximate it by a discrete timerepresentation, with input X(z), white additive noise N(z), and output Y (z):

X(z)

z−1

Y(z)

N(z)

H(z)−

1 bit ADC

1 bit DAC

• it is easy to verify that Y (z) = X(z) + (1− z−1)N(z);

• this means that the output contains the desired signal plus noise with a highpassPSD.

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Extracting High Bit Resolution

• The quantisation noise before and after noise shaping:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40

−30

−20

−10

0

10P

SD

/ [d

B]

normalised angular frequency Ω / π

AWGNnoise shaping

(PSD of N(z) is solid blue, noise component in Y (z) is dashed red)

• if we apply this in combination with oversampling, due to the reduced noise powerin the baseband, substantial resolution can be gained.

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Summary

• We have considered multirate signal processing components, which permit tochange the sampling rate;

• oversampling ADCs have been considered as an application; although not consideredhere, oversampling can be similarly exploited to reduce the cost and increase theperformance of DACs;

• Additional benefit can be gained by noise shaping in combination with oversampling;ultimately, a singe bit converter can provide multi-bit quality (assuing that the digitalpostprocessing, e.g. anti-alias filtering) is performed at a higher bit resolution.

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Makalah Elektro - (Digital) Signal Processing

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