ALARI/DSP INTRODUCTION-1

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

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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

INTRODUCTION-1 : Course schedule
Monday 14h h00: Introduction – Questions & Answers (Toon van Waterschoot) Tuesday 9h00 – 10h30: Lecture-1 (Marc Moonen) 11h h00: Exercise Session-1 (Toon van Waterschoot) 14h00 – 15h30: Lecture-2 (Marc Moonen) 16h h00: Exercise Session-2 (Toon van Waterschoot) Wednesday 9h00 – 10h30: Lecture-3 (Marc Moonen) 11h h00: Exercise Session-3 (Toon van Waterschoot) 14h00 – 15h30: Lecture-4 (Marc Moonen) 16h h00: Exercise Session-4 (Toon van Waterschoot) Thursday 9h00 – 10h30: Lecture-5 (Marc Moonen) 11h h00: Exercise Session-5 (Toon van Waterschoot) 14h00 – 15h30: Lecture-6 (Marc Moonen) 16h h00: Exercise Session-6 (Toon van Waterschoot) Friday 9h00 – 10h30: Lecture-7 (Marc Moonen) 11h h00: Exercise Session-7 (Toon van Waterschoot) 14h00 – 15h30: Lecture-8 (Marc Moonen) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

INTRODUCTION-1 : Course webpage
Toon van Waterschoot & Marc Moonen INTRODUCTION-1

INTRODUCTION-1 : Overview
Discrete-time signals sampling, quantization, reconstruction Stochastic signal theory deterministic & random signals, (auto-)correlation functions, power spectra, … Discrete-time systems LTI, impulse response, FIR/IIR, causality & stability, convolution & filtering, … Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

z-transform and Fourier transform region of convergence, causality & stability, properties, frequency spectrum, transfer function, pole-zero representation, … Elementary digital filters shelving filters, presence filters, all-pass filters Discrete transforms DFT, FFT, properties, fast convolution, overlap-add/overlap-save, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Introduction: 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, … here… Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Introduction: analog vs. digital SP
Analog signal processing: “how things used to be” Analog world Analog electrical signal processing circuit Analog IN Analog OUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Digital signal processing in the analog world Analog world Digital world Analog world Analog-to- digital conversion Digital-to- analog conversion DSP Analog IN Digital IN Digital OUT Analog OUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Analog world Analog input: microphone voltage, satellite receiver voltage, … Analog output: loudspeaker voltage, antenna voltage, … VIN VOUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Digital signal processing in the analog world Analog world Digital world Analog world Analog-to- digital conversion Digital-to- analog conversion DSP Analog IN Digital IN Digital OUT Analog OUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Introduction: design example
Goal: design and implement an audio effect which mixes a scaled and delayed version of an audio signal to the original signal Example: design of a “delay” audio effect mixing operation Analog IN Analog OUT scaling operation delay operation Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Example: design of a “delay” audio effect Analog design: mixing operation Analog IN Analog OUT delay operation scaling operation Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Example: design of a “delay” audio effect Digital design: x[k] y[k] y[k] = x[k] + K*y[k-D] mixing operation ADC DAC Analog IN Analog OUT delay operation write new sample  buffer = {y[k], y[k-1], … y[k-D]} read delayed sample inside the DSP scaling operation K*y[k-D] Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Example: design of a “delay” audio effect 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time signals: sampling-quantization
Analog signal processing Joseph Fourier ( ) Analog Domain (Continuous-Time Domain) Analog Signal Processing Circuit Analog IN Analog OUT (=Spectrum/Fourier Transform) Toon van Waterschoot & Marc Moonen INTRODUCTION-1 22

Discrete-time signals: sampling-quantization
Analog world Digital world Analog world Analog-to- digital conversion Digital-to- analog conversion DSP Analog IN Digital IN Digital OUT Analog OUT sampling quantization Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time signals: sampling
time-domain sampling amplitude amplitude discrete-time [k] continuous-time signal discrete-time signal impulse train continuous-time (t) 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 = Toon van Waterschoot & Marc Moonen INTRODUCTION-1

spectrum replication time domain: frequency domain: magnitude magnitude frequency (f) frequency (f) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Harry Nyquist (7 februari 1889 – 4 april 1976) 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time signals: quantization
B-bit quantization quantized discrete-time signal =digital signal discrete-time signal amplitude discrete time [k] amplitude discrete time [k] Q 2Q 3Q -Q -2Q -3Q R Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time signals: reconstruction
Analog world Digital world Analog world Analog-to- digital conversion Digital-to- analog conversion DSP Analog IN Digital IN Digital OUT Analog OUT reconstruction Toon van Waterschoot & Marc Moonen INTRODUCTION-1

reconstructor: ‘fill the gaps’ between adjacent samples e.g. staircase reconstructor: amplitude discrete time [k] amplitude continuous time (t) discrete-time/digital signal reconstructed analog signal Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

anti-image postfilter: low-pass filter to remove spurious high frequency components due to imperfect reconstruction comparable to the anti-aliasing prefilter Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time signals: conclusion
DSP system block scheme: Digital OUT x(t) Analog IN DSP Digital IN sampler quantizer anti-aliasing prefilter anti-image postfilter reconstructor Analog OUT xp(t) x[k] xQ[k] y[k] yR(t) y(t) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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: Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Stochastic signal theory: 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 power time frequency Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time systems: introduction
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 … DSP sampler quantizer anti-aliasing prefilter anti-image postfilter reconstructor xp(t) x[k] xQ[k] y[k] yR(t) y(t) x(t) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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: input system output Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time systems: LTI systems
Linear time-invariant (LTI) systems: linearity: time-invariance: Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 time 1 amplitude the impulse response length – 1 is equal to the order of the system Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Impulse response: if the impulse response is known, the system response to an arbitrary input signal can be calculated amplitude time 1 1 1 1 = + + 1 time 1 1 1 = + + Toon van Waterschoot & Marc Moonen INTRODUCTION-1

FIR/IIR: FIR: finite impulse response IIR: infinite impulse response amplitude 1 time amplitude 1 time Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 amplitude amplitude 1 1 time time Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Discrete-time systems: convolution
amplitude 1 1 1 = + + 1 time 1 time 1 1 1 = + + Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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: Toon van Waterschoot & Marc Moonen INTRODUCTION-1

if we define: the impulse response length (with the system order) the input sequence length then the output sequence has length Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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: Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Complex number theory: complex numbers
“imaginary” roots of a polynomial equation Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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) x p(x) x p(x) x Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

fundamental theorem of algebra: every n-th order polynomial has exactly n complex roots Toon van Waterschoot & Marc Moonen INTRODUCTION-1

complex conjugate: modulus: argument: the complex numbers form a field, and all algebraic rules for real numbers also apply for complex numbers Toon van Waterschoot & Marc Moonen INTRODUCTION-1

complex plane: the modulus and argument naturally lead to a radial representation in the complex plane Im Re complex plane Toon van Waterschoot & Marc Moonen INTRODUCTION-1

Complex number theory: complex sinusoids
complex variable  complex sinusoid: from the radial representation we obtain replacing using Euler’s identity we get Toon van Waterschoot & Marc Moonen INTRODUCTION-1

circular motion: a complex sinusoid can be seen as a vector which describes a circular trajectory in the z-plane Im z-plane Re Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Im Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

z-transform and Fourier transform region of convergence, causality & stability, properties, frequency spectrum, transfer function, pole-zero representation, … Elementary digital filters shelving filters, presence filters, all-pass filters Discrete transforms DFT, FFT, properties, fast convolution, overlap-add/overlap-save, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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