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ALARI/DSP INTRODUCTION -1

ALARI/DSP INTRODUCTION -1. Toon van Waterschoot & Marc Moonen Dept. E.E./ESAT, K.U.Leuven toon.vanwaterschoot@esat.kuleuven.be http://homes.esat.kuleuven.be/~tvanwate. INTRODUCTION-1 : Who we are. KU Leuven, Belgium

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ALARI/DSP INTRODUCTION -1

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  1. ALARI/DSPINTRODUCTION-1 Toon van Waterschoot & Marc Moonen Dept. E.E./ESAT, K.U.Leuven toon.vanwaterschoot@esat.kuleuven.be http://homes.esat.kuleuven.be/~tvanwate

  2. 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

  3. 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) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  4. INTRODUCTION-1 : Course webpage Course webpage: http://homes.esat.kuleuven.be/~tvanwate/alari.html Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  5. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  6. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  7. 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

  8. 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… Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  9. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  10. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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

  12. Introduction: analog vs. digital SP Digital signal processing in the analog world Analog world Digital world Analog world Digital-to- analog conversion Analog-to- digital conversion 0110100101 1001100010 DSP Analog IN Digital IN Digital OUT Analog OUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  13. Introduction: analog vs. digital SP • Analog world • Analog input: microphone voltage, satellite receiver voltage, … • Analog output: loudspeaker voltage, antenna voltage, … VIN 0 VOUT 0 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  14. Introduction: analog vs. digital SP Digital signal processing in the analog world Analog world Digital world Analog world Digital-to- analog conversion Analog-to- digital conversion 0110100101 1001100010 DSP Analog IN Digital IN Digital OUT Analog OUT Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  15. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  16. 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

  17. Introduction: design example 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

  18. Introduction: design example Example: design of a “delay” audio effect • Digital design: y[k] x[k] mixing operation y[k] = x[k] + K*y[k-D] 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

  19. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  20. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  21. 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

  22. Discrete-time signals: sampling-quantization Analogsignal processing Joseph Fourier (1768-1830) Analog Domain (Continuous-Time Domain) Analog Signal Processing Circuit Analog IN Analog OUT (=Spectrum/Fourier Transform) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  23. Discrete-time signals: sampling-quantization Analog world Digital world Analog world Digital-to- analog conversion Analog-to- digital conversion 0110100101 1001100010 DSP Analog IN Digital IN Digital OUT Analog OUT sampling quantization Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  24. amplitude impulse train discrete-time [k] Discrete-time signals: sampling • time-domain sampling amplitude continuous-time signal discrete-time signal 0 1 2 3 4 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

  25. Discrete-time signals: sampling • spectrum replication • time domain: • frequency domain: magnitude magnitude frequency (f) frequency (f) Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  26. Discrete-time signals: sampling • sampling theorem • the analog signal spectrum has a bandwidth of fmaxHz • 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

  27. Discrete-time signals: sampling • sampling theorem: • terminology: • sampling frequency/rate fs • Nyquist frequency fs/2 • sampling interval/period Ts • e.g. CD audio: fmax¼ 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

  28. amplitude amplitude 3Q 2Q Q R 0 -Q -2Q -3Q discrete time [k] discrete time [k] Discrete-time signals: quantization • B-bit quantization quantized discrete-time signal =digital signal discrete-time signal Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  29. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  30. 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 fmax¼ 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

  31. Discrete-time signals: reconstruction Analog world Digital world Analog world Digital-to- analog conversion Analog-to- digital conversion 0110100101 1001100010 DSP Analog IN Digital IN Digital OUT Analog OUT reconstruction Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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

  33. magnitude magnitude magnitude magnitude frequency frequency frequency frequency Discrete-time signals: reconstruction • ideal reconstructor: • ideal (rectangular) low-pass filter • no distortion • staircase reconstructor: • sync-like low-pass filter • with sidelobes • distortion due to • spurious high • frequencies Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  34. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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

  36. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  37. 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

  38. 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

  39. 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

  40. 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 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  41. 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: Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  42. 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 power 0 0 time frequency Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  43. 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, … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  44. 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

  45. x(t) y(t) xp(t) x[k] xQ[k] y[k] yR(t) anti-aliasing prefilter anti-image postfilter DSP sampler quantizer reconstructor 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 • … Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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

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

  48. amplitude amplitude 1 1 0 time 0 time 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 • the impulse response length – 1 is equal to the order of the system Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  49. amplitude 1 1 1 1 0 time 0 0 0 1 1 1 1 1 0 time 0 0 0 Discrete-time systems: LTI systems • Impulse response: • if the impulse response is known, the system response to an arbitrary input signal can be calculated = + + = + + Toon van Waterschoot & Marc Moonen INTRODUCTION-1

  50. Discrete-time systems: LTI systems • FIR/IIR: • FIR: finite impulse response • IIR: infinite impulse response amplitude 1 time 0 amplitude 1 time 0 Toon van Waterschoot & Marc Moonen INTRODUCTION-1

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