P. 1 DSP-II ALARI/DSP INTRODUCTION-1 Toon van Waterschoot & Marc Moonen Dept. E.E./ESAT, K.U.Leuven...

p. 1DSP-II

ALARI/DSP

INTRODUCTION-1

Toon van Waterschoot & Marc Moonen

Dept. E.E./ESAT, K.U.Leuven

[email protected]

http://homes.esat.kuleuven.be/~tvanwate

ALARI/DSP May 2013 p. 2 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

INTRODUCTION-1 : Who we are

• KU Leuven, Belgium– Dept. of Electrical Engineering (ESAT): signal & system

theory, micro- and nano-electronics, telecommunications, electrical energy, computer & document architecture, speech and image processing, …• SCD (SISTA-COSIC-DOCARCH): system identification, signal

processing, bio-informatics, cryptography, linear algebra, …– DSP (Digital Signal Processing): digital audio and

communications• Research topics: acoustic echo and feedback

cancellation, acoustic noise reduction, dereverberation, multicarrier communication, channel equalization, …

• Applications: hearing aids, public address systems, ADSL, wireless communication systems, …


INTRODUCTION-1 : Course schedule

• Monday – 14h00 - 17h00: Introduction – Questions & Answers (Toon van Waterschoot)

• Tuesday – 9h00 – 10h30: Lecture-1 (Marc Moonen) – 11h00 - 13h00: Exercise Session-1 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-2 (Marc Moonen) – 16h00 - 19h00: Exercise Session-2 (Toon van Waterschoot)

• Wednesday – 9h00 – 10h30: Lecture-3 (Marc Moonen) – 11h00 - 13h00: Exercise Session-3 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-4 (Marc Moonen) – 16h00 - 19h00: Exercise Session-4 (Toon van Waterschoot)

• Thursday – 9h00 – 10h30: Lecture-5 (Marc Moonen) – 11h00 - 13h00: Exercise Session-5 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-6 (Marc Moonen) – 16h00 - 19h00: Exercise Session-6 (Toon van Waterschoot)

• Friday – 9h00 – 10h30: Lecture-7 (Marc Moonen) – 11h00 - 13h00: Exercise Session-7 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-8 (Marc Moonen)


Course webpage:

http://homes.esat.kuleuven.be/~tvanwate/alari.html

INTRODUCTION-1 : Course webpage


INTRODUCTION-1 : Overview

• Introduction• Discrete-time signals

sampling, quantization, reconstruction• Stochastic signal theory

deterministic & random signals, (auto-)correlation functions, power spectra, …

• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,

convolution & filtering, …• Complex number theory

complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …



• z-transform and Fourier transformregion of convergence, causality & stability, properties,

frequency spectrum, transfer function, pole-zero representation, …

• Elementary digital filtersshelving filters, presence filters, all-pass filters

• Discrete transformsDFT, FFT, properties, fast convolution,

overlap-add/overlap-save, …


Introduction: overview

• Digital signal processing?• Analog vs. digital signal processing• Example: design of a delay audio effect

– in the analog world– in the digital world


Introduction: digital signal processing?

Digital signal processing?

• Signal: a physical quantity which varies as a function of some independent variable(s)– 1-dimensional: sound signal (mechanical/electrical),

electromagnetic signal (wired/wireless), chemical concentration, …

– 2-dimensional: image– …– N-dimensional: …

• Independent variable: time, position, frequency, …

her

e…




• Processing: altering the signal characteristics to improve signal quality– equalization: to undo the (frequency-selective) effect

of passing the signal through a system (channel)– noise reduction: to remove noise/interference– signal separation: to separate multiple signals which

are present in one measurement– modulation: to prepare a signal for being transmitted

through a frequency-selective channel– …

• Processing ~ Filtering




• Digital: the signal processing is performed by a finite number of operations using a finite number of digits– discretization of independent variable: the signal is

sampled w.r.t. the (continuous) independent variable (e.g., discrete time, discrete frequency, …)

– discretization of signal value: the signal value (amplitude) is approximated on a discrete scale (quantization)

• Bits: digital signals are often represented using binary digits = bits


Introduction: analog vs. digital SP

Analog electrical

signal processing

circuit

Analog world

Analog signal processing: “how things used to be”

Analog IN Analog OUT



Analog-to-

digital

conversionDSP

Digital-to-

analog

conversion

Analog world Analog worldDigital world

Analog IN Analog OUTDigital IN

Digital OUT

0110100101

1001100010

Digital signal processing in the analog world



• Analog world– Analog input: microphone voltage, satellite receiver

voltage, …

– Analog output: loudspeaker voltage, antenna voltage, …

VIN

0

VOUT

0



Analog-to-

digital

conversionDSP

Digital-to-

analog

conversion



Digital OUT

0110100101

1001100010

Digital signal processing in the analog world



• Digital world– Digital signal processor (DSP): microprocessor

designed particularly for signal processing operations, incorporated in sound card, modem, mobile phone, mp3 player, digital camera, digital tv, hearing aid, …


Introduction: design example

Example: design of a “delay” audio effect

• Goal: design and implement an audio effect which mixes a scaled and delayed version of an audio signal to the original signal

delay

operation


scaling

operation

mixing

operation



Analog design:



mixing

operation

delay

operationscaling

operation



Digital design:


Analog IN Analog OUTADC DAC

write new sample

buffer = {y[k], y[k-1], … y[k-D]}

read delayed sample

K*y[k-D]

y[k] = x[k] + K*y[k-D]

x[k]

inside

the DSP

mixing

operation

delay

operationscaling

operation

y[k]




• Analog design:– design of analog circuits– manufacturing of print board– assembly of analog components

• Digital design:– design of digital algorithm– compilation on digital signal processor

circuit design algorithm design

application-specific hardware re-usable hardware


Discrete-time signals: overview

• A/D conversion: sampling and quantization– time-domain sampling & spectrum replication– sampling theorem– anti-aliasing prefilters– quantization– oversampling and noise shaping

• D/A conversion: reconstruction– ideal vs. realistic reconstructors– anti-image postfilters

• Conclusion: DSP system block scheme

ALARI/DSP May 2013 p. 22

Discrete-time signals: sampling-quantization

Analog Signal

Processing Circuit

Analog Domain (Continuous-Time Domain)

Analog signal processing


)(tu

dtetufU tfj ..2).()(

(=Spectrum/Fourier Transform)

)(ty

dtetyfY tfj ..2).()(

Jose

ph

Fo

uri

er (

1768

-183

0)

Toon van Waterschoot & Marc Moonen INTRODUCTION-1


Discrete-time signals: sampling-quantization

Analog-to-

digital

conversionDSP

Digital-to-

analog

conversion



Digital OUT

0110100101

1001100010

samplingquantization


Discrete-time signals: sampling

k

sD Tkttxtx ).().()(

• time-domain samplingamplitude amplitude

discrete-time [k]continuous-time (t)

impulse train

).(][ sTkxkx )(tx

sT

It will turn out (page 27) that a spectrum can be computed from x[k], which (remarkably) will be equal to the spectrum (Fourier transform) of the (continuous-time) sequence of impulses =

discrete-time

signal

continuous-time

signal

0 1 2 3 4



• spectrum replication– time domain:

– frequency domain:

magnitude

frequency (f)

magnitude

frequency (f)

k

sD Tkttxtx ).().()(

k ss

D T

kfX

TfX )(.

1)(

)( fX )( fX D



• sampling theorem– the analog signal spectrum has a bandwidth of fmax Hz

– the spectrum replicas are separated with fs =1/Ts Hz

– no spectral overlap if and only if

magnitude

frequency

ALARI/DSP May 2013 p. 27

• sampling theorem:– terminology:

• sampling frequency/rate fs

• Nyquist frequency fs/2

• sampling interval/period Ts

– e.g. CD audio: fmax ¼ 20 kHz ) fs = 44,1 kHz

Toon van Waterschoot & Marc Moonen INTRODUCTION-1


• anti-aliasing prefilters:– if then frequencies above the Nyquist

frequency will be ‘folded back’ to lower frequencies

= aliasing– to avoid aliasing, the sampling operation is usually

preceded by a low-pass anti-aliasing filter

Har

ry N

yqui

st (

7 fe

brua

ri 18

89 –

4 a

pril

1976

)


Discrete-time signals: quantization

• B-bit quantization

amplitude

discrete time [k]

0Q

2Q3Q

-Q-2Q-3Q

R

)1on width quantizati

range(log bits ofnumber 2

Q

RB

amplitude

discrete time [k]

quantized discrete-time signal

=digital signaldiscrete-time signal

][kx ][kxQ



• B-bit quantization:– the quantization error can

only take on values between and– hence can be considered as a random noise

signal with range – the signal-to-noise ratio (SNR) of the B-bit quantizer can

then be defined as the ratio of the signal range and the quantization noise range :

= the “6dB per bit” rule



• oversampling:– it is possible to make a trade-off between sampling rate

and quantization noise– using a ‘coarse’ quantizer may be compensated by

sampling at a higher rate = oversampling– e.g. an increasing number of audio recordings is done at

a sampling rate of 96 kHz (while fmax ¼ 20 kHz )

• noise shaping:– the quantization noise is typically assumed to be white– the noise spectrum may be altered to decrease its

disturbing effect = noise shaping– e.g. psycho-acoustic noise shaping in audio quantizing


Discrete-time signals: reconstruction

Analog-to-

digital

conversionDSP

Digital-to-

analog

conversion



Digital OUT

0110100101

1001100010

reconstruction



• reconstructor: – ‘fill the gaps’ between adjacent samples– e.g. staircase reconstructor:

amplitude

discrete time [k]

amplitude

continuous time (t)

reconstructed

analog signal

discrete-time/digital signal

][kx )(txR



• ideal reconstructor:– ideal (rectangular) low-pass filter– no distortion

magnitude

frequency

magnitude

frequency

• staircase reconstructor:– sync-like low-pass filter with sidelobes– distortion due to spurious high frequencies

magnitude

frequency

magnitude

frequency

)( fX D

)( fX D



• anti-image postfilter:– low-pass filter to remove spurious high frequency

components due to imperfect reconstruction– comparable to the anti-aliasing prefilter


Discrete-time signals: conclusion

Digital OUT

x(t)

Analog IN

DSPDigital

IN

sampler quantizeranti-

aliasing prefilter

anti-image

postfilterreconstructor

Analog OUT

xp(t) x[k] xQ[k] y[k] yR(t)

y(t)

DSP system block scheme:


Stochastic signal theory: overview

• Signal types:– deterministic signals– random signals

• Correlation functions and power spectra:– autocorrelation function & power spectrum– cross-correlation function & cross-spectrum– (joint) wide sense stationarity

• White noise:– Gaussian white noise– uniform white noise


Stochastic signal theory: signal types

• Deterministic signals– a deterministic signal is an explicit function of time,

e.g. • Random signals

– a random signal is ‘unpredictable’ in a sense– some information on the signal behaviour may be

available, e.g.• probability density function (PDF) • mean • variance• autocorrelation function• …


Stochastic signal theory: corr/spectra

• Autocorrelation function– measure of the dependence between successive

samples (with lag ) of a random signal

• Power spectrum– measure of the frequency content of a random signal– Fourier transform of the autocorrelation function



• Cross-correlation function– measure of the dependence between successive

samples (with lag ) of two different random signals

• Cross-spectrum– measure of spectral overlap between two random signals– Fourier transform of the cross-correlation function



• Wide-sense stationarity (WSS):– a random signal is wide-sense stationary if its mean and

autocorrelation function are independent of time:

• Joint wide-sense stationarity (joint WSS)– two random signals are jointly wide-sense stationary if

their cross-correlation function is independent of time:


Stochastic signal theory: white noise

• White noise:– a zero-mean white noise signal has an impulse

autocorrelation function and a flat power spectrum:

– Gaussian white noise has a Gaussian PDF (Matlab function randn)

– uniform white noise has a uniform PDF (Matlab function rand)

power

time0

power

frequency0


Discrete-time systems: overview

• Introduction:– discrete-time systems– I/O behaviour

• LTI systems:– linear time-invariant systems– impulse response– FIR/IIR– causality– stability

• Convolution:– direct form– matrix form


Discrete-time systems: introduction

• discrete-time systems:– any system implemented on a digital signal processor:

– discrete-time model of continuous-time system, e.g.• wireless channel in mobile communications• twisted pair telephone line• acoustic echo channel between loudspeaker and microphone• …

DSPsampler quantizeranti-

aliasing prefilter

anti-image

postfilterreconstructor

xp(t) x[k] xQ[k] y[k] yR(t)

y(t)x(t)


Discrete-time systems: introduction

• input/output (I/O) behaviour:– mapping of input sequence on output sequence:

– the output signal is a function of the input signal:

systeminput output


Discrete-time systems: LTI systems

• Linear time-invariant (LTI) systems:– linearity:

– time-invariance:



• Impulse response:– LTI systems are characterized uniquely by their impulse

response = the system output in response to a unit impulse input signal

amplitude

time

1

0time

1

amplitude

0

– the impulse response length – 1 is equal to the order of the system



• Impulse response:– if the impulse response is known, the system response

to an arbitrary input signal can be calculated

amplitude

time

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

time

= + +

+ +=



• FIR/IIR:– FIR: finite impulse response

– IIR: infinite impulse response

time

1

amplitude

0

time

1

amplitude

0



• Causality:– a causal system has an impulse response that is zero

for all negative time indices– a non-causal system has an impulse response that has

some non-zero coefficients on the negative time axis, i.e. the system output depends on future input values

time

1

amplitude

0 time

1

amplitude

0



• Stability:– a system is said to be stable if a bounded input signal

always generates a bounded output signal:

– a necessary and sufficient condition for stability is that the system impulse response be absolutely summable:

– instability can only occur with IIR systems


Discrete-time systems: convolution

amplitude

time

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

time

= + +

+ +=



• Convolution:– the expression

can be written in a more general form:

– this operation is called convolution of the system impulse response with the input signal

– shorthand notation:



• Convolution:

– if we define:• the impulse response length (with the system

order)• the input sequence length

– then the output sequence has length



• Direct form convolution:– one way to perform the convolution of and

is to directly calculate the summation

– this is done for all time indices – an appropriate choice for the summation limits is:



• Matrix form convolution:– another way to perform the convolution of and

is by rewriting the summation as a matrix product

– the signal vectors and the impulse response matrix are defined as follows (with e.g. M=2 and L=4)


Complex number theory: overview

• Complex numbers:– roots of a quadratic polynomial equation– fundamental theorem of algebra– complex numbers– complex plane

• Complex sinusoids– complex numbers complex sinusoids– circular motion– positive and negative frequencies– sinusoidal motion


Complex number theory: complex numbers

complex numbers?

“imaginary” roots of a polynomial equation



• roots of a quadratic polynomial equation:– consider a quadratic polynomial, describing a parabola:

– the roots of the polynomial correspond to the points where the parabola crosses the horizontal -axis



• roots of a quadratic polynomial equation:– if the polynomial

has 2 real roots, and the parabola has 2 distinct intercepts with the -axis

– if the polynomial has 1 real root (with multiplicity 2), and the parabola has 1 intercept (tangent point) with the -axis

– if the polynomial has no real roots, and the parabola has no intercepts with the -axis

p(x)

xp(x)

xp(x)

x



• roots of a quadratic polynomial equation:– alternatively, if we could say that

the polynomial has 2 “imaginary roots”, and the parabola has 2 “imaginary” intercepts with the -axis

– these imaginary roots are represented as complex numbers:

with

p(x)

x



fundamental theorem of algebra:

every n-th order polynomial has exactly n complex roots



• complex numbers:– complex number:

– complex conjugate:

– modulus:

– argument:

• the complex numbers form a field, and all algebraic rules for real numbers also apply for complex numbers



• complex plane:– the modulus and argument naturally lead to a radial

representation in the complex plane

Im

Re

complex

plane


Complex number theory: complex sinusoids

• complex variable complex sinusoid:

– from the radial representation we obtain

– replacing

– using Euler’s identity we get



• circular motion:– a complex sinusoid can be seen as a vector which

describes a circular trajectory in the z-plane

Im

Re

z-plane



• positive and negative frequencies:– for positive frequencies the circular motion is in

counterclockwise direction– for negative frequencies the circular motion is in

clockwise direction

Im

Re

Im

Re



• sinusoidal motion:– sinusoidal motion is the projection of circular motion

onto any straight line in the z-plane, e.g.,• is the projection of onto the Re-axis• is the projection of onto the Im-axis

Im

Re



• z-transform and Fourier transformregion of convergence, causality & stability, properties,

frequency spectrum, transfer function, pole-zero representation, …

• Elementary digital filtersshelving filters, presence filters, all-pass filters

• Discrete transformsDFT, FFT, properties, fast convolution,

overlap-add/overlap-save, …

P. 1 DSP-II ALARI/DSP INTRODUCTION-1 Toon van Waterschoot & Marc Moonen Dept. E.E./ESAT, K.U.Leuven...

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