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

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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, …

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ALARI/DSP May 2013 p. 3 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)

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ALARI/DSP May 2013 p. 4 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Course webpage: INTRODUCTION-1 : Course webpage

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ALARI/DSP May 2013 p. 5 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-1 : Overview Introduction 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, …

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ALARI/DSP May 2013 p. 6 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-2 : Overview 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, …

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ALARI/DSP May 2013 p. 7 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

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ALARI/DSP May 2013 p. 8 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 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, … here…

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ALARI/DSP May 2013 p. 9 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: digital signal processing? Digital signal processing? 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

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ALARI/DSP May 2013 p. 10 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: digital signal processing? Digital signal processing? 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

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ALARI/DSP May 2013 p. 11 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: analog vs. digital SP Analog electrical signal processing circuit Analog world Analog signal processing: “how things used to be” Analog INAnalog OUT

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ALARI/DSP May 2013 p. 12 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: analog vs. digital SP Analog-to- digital conversion DSP Digital-to- analog conversion Analog world Digital world Analog INAnalog OUTDigital IN Digital OUT Digital signal processing in the analog world

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ALARI/DSP May 2013 p. 13 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: analog vs. digital SP Analog world –Analog input: microphone voltage, satellite receiver voltage, … –Analog output: loudspeaker voltage, antenna voltage, … V IN 0 V OUT 0

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ALARI/DSP May 2013 p. 14 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: analog vs. digital SP Analog-to- digital conversion DSP Digital-to- analog conversion Analog world Digital world Analog INAnalog OUTDigital IN Digital OUT Digital signal processing in the analog world

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ALARI/DSP May 2013 p. 15 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: analog vs. digital SP 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, …

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ALARI/DSP May 2013 p. 16 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 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 Analog INAnalog OUT scaling operation mixing operation

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ALARI/DSP May 2013 p. 17 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: design example Analog design: Example: design of a “delay” audio effect Analog INAnalog OUT mixing operation delay operation scaling operation

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ALARI/DSP May 2013 p. 18 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: design example Digital design: Example: design of a “delay” audio effect Analog INAnalog OUT ADCDAC 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 operation scaling operation y[k]

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ALARI/DSP May 2013 p. 19 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Introduction: design example 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

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ALARI/DSP May 2013 p. 20 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-1 : Overview Introduction 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, …

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ALARI/DSP May 2013 p. 21 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

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ALARI/DSP May 2013 p. 22 Discrete-time signals: sampling-quantization Analog Signal Processing Circuit Analog Domain (Continuous-Time Domain) Analog signal processing Analog INAnalog OUT (=Spectrum/Fourier Transform) Joseph Fourier ( ) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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ALARI/DSP May 2013 p. 23 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: sampling-quantization Analog-to- digital conversion DSP Digital-to- analog conversion Analog world Digital world Analog INAnalog OUTDigital IN Digital OUT sampling quantization

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ALARI/DSP May 2013 p. 24 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: sampling time-domain sampling amplitude discrete-time [k] continuous-time (t) impulse train 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

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ALARI/DSP May 2013 p. 25 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: sampling spectrum replication –time domain: –frequency domain: magnitude frequency (f) magnitude frequency (f)

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ALARI/DSP May 2013 p. 26 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: sampling sampling theorem –the analog signal spectrum has a bandwidth of f max Hz –the spectrum replicas are separated with f s =1/T s Hz –no spectral overlap if and only if magnitude frequency

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ALARI/DSP May 2013 p. 27 sampling theorem: –terminology: sampling frequency/rate f s Nyquist frequency f s /2 sampling interval/period T s –e.g. CD audio: f max ¼ 20 kHz ) f s = 44,1 kHz Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: sampling anti-aliasing prefilters: –ifthen 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 Harry Nyquist (7 februari 1889 – 4 april 1976)

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ALARI/DSP May 2013 p. 28 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: quantization B-bit quantization amplitude discrete time [k] 0 Q 2Q2Q 3Q3Q -Q-Q -2Q -3Q R amplitude discrete time [k] quantized discrete-time signal =digital signal discrete-time signal

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ALARI/DSP May 2013 p. 29 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: quantization 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

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ALARI/DSP May 2013 p. 30 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: quantization 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 f max 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

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ALARI/DSP May 2013 p. 31 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: reconstruction Analog-to- digital conversion DSP Digital-to- analog conversion Analog world Digital world Analog INAnalog OUTDigital IN Digital OUT reconstruction

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ALARI/DSP May 2013 p. 32 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: reconstruction reconstructor: –‘fill the gaps’ between adjacent samples –e.g. staircase reconstructor: amplitudediscrete time [k]amplitudecontinuous time (t) reconstructed analog signal discrete-time/digital signal

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ALARI/DSP May 2013 p. 33 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: reconstruction 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

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ALARI/DSP May 2013 p. 34 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: reconstruction anti-image postfilter: –low-pass filter to remove spurious high frequency components due to imperfect reconstruction –comparable to the anti-aliasing prefilter

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ALARI/DSP May 2013 p. 35 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time signals: conclusion Digital OUT x(t) Analog IN DSP Digital IN samplerquantizer anti- aliasing prefilter anti- image postfilter reconstructor Analog OUT x p (t)x[k]x Q [k]y[k]y R (t) y(t) DSP system block scheme:

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ALARI/DSP May 2013 p. 36 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-1 : Overview Introduction 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, …

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ALARI/DSP May 2013 p. 37 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

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ALARI/DSP May 2013 p. 38 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 …

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ALARI/DSP May 2013 p. 39 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

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ALARI/DSP May 2013 p. 40 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Stochastic signal theory: corr/spectra 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

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ALARI/DSP May 2013 p. 41 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Stochastic signal theory: corr/spectra 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:

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ALARI/DSP May 2013 p. 42 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 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 time 0 power frequency 0

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ALARI/DSP May 2013 p. 43 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-1 : Overview Introduction 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, …

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ALARI/DSP May 2013 p. 44 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

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ALARI/DSP May 2013 p. 45 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 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 … DSP samplerquantizer anti- aliasing prefilter anti- image postfilter reconstructor x p (t)x[k]x Q [k]y[k]y R (t) y(t)x(t)

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ALARI/DSP May 2013 p. 46 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: system input output

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ALARI/DSP May 2013 p. 47 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems Linear time-invariant (LTI) systems: –linearity: –time-invariance:

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ALARI/DSP May 2013 p. 48 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems Impulse response: –LTI systems are characterized uniquely by their impulse response = the system output in response to a unit impulse input signal amplitude time amplitude 0 –the impulse response length – 1 is equal to the order of the system

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ALARI/DSP May 2013 p. 49 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems Impulse response: –if the impulse response is known, the system response to an arbitrary input signal can be calculated amplitude time =++ ++=

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ALARI/DSP May 2013 p. 50 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems FIR/IIR: –FIR: finite impulse response –IIR: infinite impulse response time 1 amplitude 0 time 1 amplitude 0

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ALARI/DSP May 2013 p. 51 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems 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 0time 1 amplitude 0

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ALARI/DSP May 2013 p. 52 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: LTI systems 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

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ALARI/DSP May 2013 p. 53 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: convolution amplitude time =++ ++=

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ALARI/DSP May 2013 p. 54 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: convolution 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:

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ALARI/DSP May 2013 p. 55 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: convolution Convolution: –if we define: the impulse response length (with the system order) the input sequence length –then the output sequence has length

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ALARI/DSP May 2013 p. 56 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: convolution 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:

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ALARI/DSP May 2013 p. 57 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Discrete-time systems: convolution 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)

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ALARI/DSP May 2013 p. 58 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-1 : Overview Introduction 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, …

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ALARI/DSP May 2013 p. 59 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

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ALARI/DSP May 2013 p. 60 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers complex numbers? “imaginary” roots of a polynomial equation

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ALARI/DSP May 2013 p. 61 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers 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

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ALARI/DSP May 2013 p. 62 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers 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 x x

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ALARI/DSP May 2013 p. 63 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers 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

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ALARI/DSP May 2013 p. 64 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers fundamental theorem of algebra: every n-th order polynomial has exactly n complex roots

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ALARI/DSP May 2013 p. 65 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers 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

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ALARI/DSP May 2013 p. 66 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex numbers complex plane: –the modulus and argument naturally lead to a radial representation in the complex plane Im Re complex plane

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ALARI/DSP May 2013 p. 67 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

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ALARI/DSP May 2013 p. 68 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex sinusoids circular motion: –a complex sinusoid can be seen as a vector which describes a circular trajectory in the z-plane Im Re z-plane

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ALARI/DSP May 2013 p. 69 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex sinusoids 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

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ALARI/DSP May 2013 p. 70 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 Complex number theory: complex sinusoids 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

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ALARI/DSP May 2013 p. 71 Toon van Waterschoot & Marc Moonen INTRODUCTION-1 INTRODUCTION-2 : Overview 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, …

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