Basics on Digital Signal Processing


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1 Basics on Digital Signal Processing Introduction Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras
2 Outline of the Course 1. Introduction (sampling quantization) 2. Signals and Systems 3. ZTransform 4. The Discreet and the Fast Fourier Transform 5. Linear Filter Design 6. Noise 7. Median Filters 2/36
3 Voltage [V] Voltage [V] Analog & digital signals Analog Continuous function V of continuous variable t (time, space etc) : V(t). Sampled Signal Digital Discrete function V k of discrete sampling variable t k, with k = integer: V k = V(t k ) time [ms] Uniform (periodic) sampling. Sampling frequency f S = 1/ t S t s t s sampling time, t k [ms] 3/36
4 Analog & digital systems 4/36
5 Digital vs analog processing Digital Signal Processing (DSPing) Advantages More flexible. Often easier system upgrade. Data easily stored memory. Better control over accuracy requirements. Reproducibility. Linear phase No drift with time and temperature Limitations A/D & signal processors speed: wideband signals still difficult to treat (realtime systems). Finite wordlength effect. 5/36
6 DSPing: aim & tools Applications Predicting a system s output. Implementing a certain processing task. Studying a certain signal. Hardware General purpose processors (GPP), controllers. Digital Signal Processors (DSP). Programmable logic ( PLD, FPGA ). Fast Faster realtime DSPing Software Programming languages: Pascal, C / C++... High level languages: Matlab, Mathcad, Mathematica Dedicated tools (ex: filter design s/w packages). 6/36
7 Related areas 7/36
8 Applications 8/36
9 Important digital signals δ(nt s ) δ[(n3)τ s ] nτ s past Unit Impulse or Unit Sample. The most important signal for two reasons δ(n)=1 for n=0 u(nt s) nτ s past Unit Step u(n)=1 for n0 δ(n)=u(n)u(n1) r(nt s) nτ s past Unit Ramp r(n)=nu(n) 9/36
10 Digital system example General scheme Sometimes steps missing  Filter + A/D (ex: economics);  D/A + filter (ex: digital output wanted). V V A A V ms ms k k Filter Antialiasing Antialiasing A/D A/D Digital Processing Digital Processing D/A ANALOG DOMAIN DIGITAL DOMAIN Topics of this lecture. V ms ms Filter Reconstruction ANALOG DOMAIN 10/36
11 Digital system implementation ANALOG INPUT KEY DECISION POINTS: Analysis bandwidth, Dynamic range Antialiasing Filter A/D Pass / stop bands. Sampling rate. No. of bits. Parameters. 1 2 Digital Processing DIGITAL OUTPUT Digital format. What to use for processing? 3 11/36
12 AD/DA Conversion General Scheme 12/36
13 AD Conversion  Details 13/36
14 Sampling 14/36
15 1 Sampling How fast must we sample a continuous signal to preserve its info content? Ex: train wheels in a movie. 25 frames (=samples) per second. Train starts wheels go clockwise. Train accelerates wheels go counterclockwise. Why? Frequency misidentification due to low sampling frequency. 15/36
16 Rotating Disk How fast do we have to instantly stare at the disk if it rotates with frequency 0.5 Hz? 16/36
17 1 The sampling theorem Theo* A signal s(t) with maximum frequency f MAX can be recovered if sampled at frequency f S > 2 f MAX. * Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel nikov. Naming gets confusing! Nyquist frequency (rate) f N = 2 f MAX or f MAX or f S,MIN or f S,MIN /2 Example s(t) 3cos(50π t) 10sin(300π t) cos(100π t) Condition on f S? F 1 F 2 F 3 F 1 =25 Hz, F 2 = 150 Hz, F 3 = 50 Hz f S > 300 Hz f MAX 17/36
18 Sampling and Spectrum 18/36
19 1 Sampling lowpass signals (a) Continuous spectrum (a) Bandlimited signal: frequencies in [B, B] (f MAX = B). (b) B 0 B f Discrete spectrum No aliasing (b) Time sampling frequency repetition. f S > 2 B no aliasing. B 0 B f S /2 f (c) Discrete spectrum Aliasing & corruption (c) f S 2 B aliasing! 0 f S /2 f Aliasing: signal ambiguity in frequency domain 19/36
20 1 Antialiasing filter (a) Out of band noise Signal of interest Out of band noise (a),(b) Outofband noise can aliase into band of interest. Filter it before! (b) (c) B 0 B f B 0 B f S /2 f (c) Antialiasing filter Passband: depends on bandwidth of interest. Attenuation A MIN : depends on ADC resolution ( number of bits N). A MIN, db ~ 6.02 N Outofband noise magnitude. Other parameters: ripple, stopband frequency... 20/36
21 1 Undersampling Using spectral replications to reduce sampling frequency f S req ments. Bandpass signal centered on f C B m 2fC B 2fC B fs m 1 m, selected so that f S > 2B 0 f C f Example f C = 20 MHz, B = 5MHz Without undersampling f S > 40 MHz. With undersampling f S = 22.5 MHz (m=1); = 17.5 MHz (m=2); = MHz (m=3). f S 0 f S 2f S f Advantages Slower ADCs / electronics needed. Simpler antialiasing filters. f C 21/36
22 Quantization and Coding N Quantization Levels q Quantization Noise 22/36
23 2 SNR of ideal ADC RMS input SNR ideal 20 log10 (1) RMS(eq) Also called SQNR (signaltoquantisationnoise ratio) Assumptions Ideal ADC: only quantisation error e q (p(e) constant, no stuck bits ) e q uncorrelated with signal. ADC performance constant in time. RMS 1 T T V FSR 2 0 input sin ωt 2 dt V FSR 2 2 Input(t) = ½ V FSR sin( t). p(e) quantisation error probability density RMS(eq) q/2 2 eq peq deq q/2 q 12 VFSR 2 N 12 1 q (sampling frequency f S = 2 f MAX ) q 2 q 2 e q Error value 23/36
24 2 SNR of ideal ADC  2 Substituting in (1) : SNR ideal 6.02N1.76 [db] (2) One additional bit SNR increased by 6 db Real SNR lower because:  Real signals have noise.  Forcing input to full scale unwise.  Real ADCs have additional noise (aperture jitter, nonlinearities etc). Actually (2) needs correction factor depending on ratio between sampling freq & Nyquist freq. Processing gain due to oversampling. 24/36
25 Coding  Conventional 25/36
26 Coding Flash AD 26/36
27 DAC process 27/36
28 Oversampling Noise shaping PSD Nyquist Sampler f b f N (a) Oversampling OSR=4 f The oversampling process takes apart the images of the signal band. (b) f s =4f N f 0 PSD Signal f N /2 Quantization noise in Nyquist converters Quantization noise in Oversampling converters f s /2 When the sampling rate increases (4 times) the quantization noise spreads over a larger region. The quantization noise power in the signal band is 4 times smaller. PSD Signal Quantization noise Nyquist converters Quantization noise Oversampling converters Quantization noise Oversampling and noise shaping converters Spectrum at the output of a noise shaping quantizer loop compared to those obtained from Nyquist and Oversampling converters. 0 F N /2 frequency F s /2 28/36
29 Digital Systems A discreettime system is a device or algorithm that operates on an input sequence according to some computational procedure It may be A general purpose computer A microprocessor dedicated hardware A combination of all these 29/36
30 Linear, Time Invariant Systems System Properties linear Time Invariant Stable Causal y( n) N k0 a k x( n k) Convolution 30/36
31 Linear Systems  Convolution 5+71=11 terms 31/36
32 Linear Systems  Convolution 5+71=11 terms 32/36
33 General Linear Structure y( n) M k0 a k x( n k) L k1 b k y( n k) 33/36
34 Simple Examples 34/36
35 Linearity Superposition Frequency Preservation Principle of Superposition x 1 (n) H y 1 (n) ax 1 (n)+bx 2 (n) H ay 1 (n)+by 2 (n) x 2 (n) H y 2 n) Principle of Superposition Frequency Preservation x 1 (n) x 2 1 (n) x 2 x 1 (n)+x 2 (n) x 2 x 1 2 (n)+x 2 2 (n)+2 x 1 (n) x 2 (n) Nonlinear x 2 (n) x 2 x 2 2 (n) If y(n)=x 2 (n) then for x(n)=sin(nω) y(n)=sin 2 (nω)= cos(2nω) 35/36
36 The END Have a nice Weekend Back on Tuesday 36/36
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