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 More flexible. • Often easier system upgrade. • Data easily stored memory. • Better control over accuracy requirements. • Reproducibility. • Linear phase • No drift with time and temperature
Basics on Digital Signal Processing Introduction Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras Outline of the Course Introduction (sampling – quantization) Signals and Systems Z-Transform The Discreet and the Fast Fourier Transform Linear Filter Design Noise Median Filters 2/36 Analog & digital signals Analog Digital Discrete function Vk of discrete sampling variable tk, with k = integer: Vk = V(tk) Sampled Signal 0.3 0.3 0.2 0.2 Voltage [V] Voltage [V] Continuous function V of continuous variable t (time, space etc) : V(t) 0.1 -0.1 -0.2 0.1 ts ts -0.1 -0.2 time [ms] 10 sampling time, tk [ms] 10 Uniform (periodic) sampling Sampling frequency fS = 1/ tS 3/36 Analog & digital systems 4/36 Digital vs analog processing Digital Signal Processing (DSPing) Advantages Limitations • Often easier system upgrade • A/D & signal processors speed: wide-band signals still difficult to treat (real-time systems) • Data easily stored -memory • Finite word-length effect • More flexible • Better control over accuracy requirements • Reproducibility • Linear phase • No drift with time and temperature 5/36 DSPing: aim & tools Applications • Predicting a system’s output • Implementing a certain processing task • Studying a certain signal • General purpose processors (GPP), -controllers Hardware Software • Digital Signal Processors (DSP) Fast • Programmable logic ( PLD, FPGA ) Faster real-time DSPing • Programming languages: Pascal, C / C++ • “High level” languages: Matlab, Mathcad, Mathematica… • Dedicated tools (ex: filter design s/w packages) 6/36 Related areas 7/36 Applications 8/36 Important digital signals Unit Impulse or Unit Sample δ(nTs) δ[(n-3)Τs] The most important signal for two reasons nΤs past u(nTs) δ(n)=1 for n=0 Unit Step u(n)=1 for n0 nΤs past δ(n)=u(n)-u(n-1) r(nTs) Unit Ramp r(n)=nu(n) nΤs past 9/36 Digital system example General scheme ms V Sometimes steps missing ms A (ex: economics); k - D/A + filter (ex: digital output wanted) Antialiasing A k V V ms A/D Digital Processing Digital Processing D/A Filter Reconstruction ms ANALOG DOMAIN Topics of this lecture A/D DIGITAL DOMAIN - Filter + A/D Filter Filter Antialiasing ANALOG DOMAIN V 10/36 Quantization and Coding N Quantization Levels q Quantization Noise 22/36 SNR of ideal ADC Assumptions RMS input (1) SNRideal 20 log10 Ideal ADC: only quantisation error eq RMS(e q ) Also called SQNR (signal-to-quantisation-noise ratio) RMS input (p(e) constant, no stuck bits…) eq uncorrelated with signal ADC performance constant in time T V VFSR sinωt dt FSR T 2 Input(t) = ½ VFSR sin( t) p(e) quantisation error probability density q/2 RMS(e q ) eq2 p eq deq -q/2 VFSR q 12 2N 12 (sampling frequency fS = fMAX) q q q eq Error value 23/36 SNR of ideal ADC - SNRideal 6.02 N 1.76 [dB] Substituting in (1) : One additional bit (2) SNR increased by 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 Coding - Conventional 25/36 Coding – Flash AD 26/36 DAC process 27/36 Oversampling – Noise shaping PSD Nyquist Sampler f fb fN The oversampling process takes apart the images of the signal band (a) Oversampling OSR=4 f fs=4fN (b) PSD Signal Quantization noise in Nyquist converters Quantization noise in Oversampling converters fN/2 PSD Signal fs/2 Quantization noise Nyquist 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 Quantization noise Oversampling converters FN/2 frequency When the sampling rate increases (4 times) the quantization noise spreads over a larger region The quantization noise power in the signal band is times smaller Fs/2 28/36 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 Linear, Time Invariant Systems System Properties • linear •Time Invariant •Stable •Causal N y ( n ) ak x ( n k ) k 0 Convolution 30/36 Linear Systems - Convolution 5+7-1=11 terms 31/36 Linear Systems - Convolution 5+7-1=11 terms 32/36 General Linear Structure M L k 0 k 1 y (n) ak x(n k ) bk y (n k ) 33/36 Simple Examples 34/36 Linearity – Superposition – Frequency Preservation Principle of Superposition x1(n) y1(n) H ax1(n)+bx2(n) ay1(n)+by2(n) H x2(n) y2n) H Principle of Superposition Frequency Preservation x12(n) x1(n) x2 x12(n)+x22(n)+2 x1(n) x2(n) x1(n)+x2(n) x2 x2(n) Non-linear x22(n) x If y(n)=x2(n) then for x(n)=sin(nω) y(n)=sin2(nω)=0.5+0.5cos(2nω) 35/36 The END Have a nice Weekend Back on Tuesday 36/36