# 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. Z-Transform 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: wide-band signals still difficult to treat (real-time systems). Finite word-length 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 real-time 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[(n-3)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(n-1) 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 counter-clockwise. 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 low-pass signals (a) Continuous spectrum (a) Band-limited 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) Out-of-band 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 Out-of-band noise magnitude. Other parameters: ripple, stopband frequency... 20/36

21 1 Under-sampling 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 under-sampling f S > 40 MHz. With under-sampling 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 (signal-to-quantisation-noise 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, non-linearities 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

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 discreet-time 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+7-1=11 terms 31/36

32 Linear Systems - Convolution 5+7-1=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) Non-linear 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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