10:15-11 am Digital signal processing

Transcription

1 1 10:15-11 am Digital signal processing Data Conversion & Sampling Sampled Data Systems Data Converters Analog to Digital converters (A/D ) Digital to Analog converters (D/A) with Zero Order Hold Signal Reconstruction Alternate sampling & reconstruction process ZOH Example: Compact Disk Aliasing & Frequency Foldover Aliasing Aliasing example: Rotating wheel in motion picture Aliasing example: sine waves Signal Amplification & Anti-aliasing (pre)filters D/A Post processing

2 2 Data Conversion & Sampling transducer R(t) conditioning: anti-alias filtering & amplification r(t) A/D data r c D/A data i i c(t) conditioning converter computer converter & amplification C(t) actuator clock Signal flow: digital signal processing continuous voltage signals: R(t) from transducers, C(t) to actuators Analog (continuous) signals R(t) conditioned to r(t) & c(t) to C(t) analog signal r(t) converted to digital signal (sequence binary numbers) ri Analog to Digital converter (A/D): r(t) ri ci converted to analog signal c(t) Digital to Analog converter (D/A): ci c(t) computer (micro-processor) executes algorithm A/D & D/A creates sampled data system stroboscopic effect: computer "sees" system only at certain times in between: computer blind to system

3 3 Sampled Data Systems Sampling time continuous r(t) ri r(ti ), i =... -2, -1, 0, 1, 2,... at discrete instants ti linear operation h(t) = α f(t) + β g(t) hi = α fi + β gi equally spaced sample instants (usual practice) ti = i Ts ;sampling period: Ts sampling frequency: fs = 1/Ts = ωs /2π Advantages digital processing clean: no electrical and/or mechanical noise added very complicated processing algorithms possible Problems / Disadvantages signal reconstruction ci c(t) imperfect: aliasing & foldover high frequency noise folded over into lower spectrum word saturation: (overflow, underflow) possible real time applications complete algorithm execution time Ts computer always lags system high A/D & D/A data resolution quantization errors bits of accuracy digital & analog systems noise incompatible

4 4 Data Converters Interface computer / environment Input/output A/D n bit binary voltages number D/A FS = full scale voltage range of converter n bit converter: 2 n possible levels Resolution: minimum voltage increment associated with least significant bit (LSB) = FS/2 n Conventions Unipolar converters (+ only) Bipolar converters (+ & -) 0 V 0 V Converter levels usually offset by least significant bit LSB/2 FS not in converter levels

5 5 Example: 2 bit A/D converter binary number volts 2 bit converter: 2 2 = 4 levels: {00, 01, 10, 11} FS = 10 volts LSB = 10 V/2 2 = 2.5 V LSB/2 = 1.25 V Binary numbers correspond to voltage "bins" FS = 10 V not in converter range: V resolution improves with more bits

6 6 Analog to Digital converters (A/D ) r(t) A/D data r i converter r(t) t/t s r i T s t/t s process: voltage converted into binary number prevalent converter types: Successive approximation Parallel (flash) converters

7 7 Digital to Analog converters (D/A) with Zero Order Hold c i D/A data c(t) converter c(t) t/t s c i process: t/t s binary numbers converted to voltages staircase approximation to curve, improves with smaller Ts prevalent converter types: Weighted current sources R-2R Ladder Network

8 8 Signal Reconstruction Question: Given f(t), band limited by ωo = 2π fo, what is minimum sampling frequency fs = 1/Ts = ωs /2π errorless reconstruction of signal? Answer: Nyquist frequency ω N = 2 ωo, from Shannon's sampling theorem A continuous time signal f(t) with a Fourier transform F(ω) that is zero outside the interval ( - ωo, ωo ) is given uniquely by its values in equidistant points if the sampling frequency is higher than 2 ωo. The continuous time signal can be computed from the sampled signal by the interpolation formula f(t) = f(i Ts) sin ω s(t - i Ts)/2 ωs(t - i Ts)/2 i= - where ωs is the sampling angular frequency in radians per second. Shannon reconstruction NOT CAUSAL requires ALL values f(i Ts ): past, present & future no good for real time

9 9 Alternate sampling & reconstruction process Oversample: sample at rates higher than ω N = 2 ωo Oversampling Rule of Thumb: set ωs 4 ωo to 20 ωo Holds: holds value f(ti ) until next value f(ti+1 ): causal but approximate Zero order hold (ZOH) f ZOH (t) = f(ti ) ti t < ti+1 requires only present value f(ti ) D/A outputs like this piecewise constant, from right jumps at right to f(ti+1 ) staircase approximation may require post-filter to smooth jumps, but extra dynamics c(t) 6 t/t s c i T s t/t s First Order Hold: from Taylor series t - ti f FOH (t) = f(ti ) + ti - [ ti-1 f(ti ) - f(ti-1 ) ] ti t < ti

10 10 ZOH Example: Compact Disk Signal coding on CD track 0.11 µm deep x 0.5 µm wide x 0.9 µm long pits pitch (pit spacing) 1.6 µm laser reflection diffraction pattern disk surface: reflected signal out of phase, cancels source 0 in pit: reflected signal in phase with source, adds constructively 1 pits bits 10 billion pits on standard CD linear recording density 43 kbit/inch Quantization: 16 bits, 2 s complement Sampling frequency: 44.1 khz Audio range to 20 khz Advertised: 4 x oversampling, minimizes aliasing problems Digital to Analog conversion R-2R D/A Sample & Hold (ZOH) Smoothing filter Analog signal out

11 11 Aliasing & Frequency Foldover Process signal to be sampled with Fourier transform F(ω) f(t) F(ω) = f(t) e - jωt dt, bandlimited by ωo t= - signal after sampling f(t) fs(t) with sampling frequency ωs, fs(t) has Fourier transform Fs(ω) = 1 Ts F(ω + k ωs) k = - Note: Components F(ω + k ωs ) form periodic array in frequency spectra Prior eqn. sums components Major problems signal reconstruction possible only if ωs 2 ωo high frequency noise folded over into usable bandwidth

12 12 Frequency Foldover F(ω) 2 ω s ω s - ω o ω o 0 ω s 2 ω s Proper sampling ( ωs > 2 ωo ) repeated spectra (periodic) separate undistorted signal reconstruction possible F(ω) 3 ω s 2 ω s ω s ω o - ω o 0 ω s 2 ω s Reducing ωs (still ωs > 2 ωo) moves repeated spectra closer together undistorted signal reconstruction still possible 3 ω s

13 13 F(ω) - ω o ω o 0 ω s 2 ω s 5 ω s 4 ω s 3 ω s 2 ω s ω s 3 ω s 4 ω s 5 ω s Further reducing ωs ( ωs = 2 ωo) moves repeated spectra still closer limit of undistorted signal reconstruction (Nyquist frequency ω N = 2 ωo ) F(ω) 6 ω s 4 ω s 2 ω s ω o - ω o 0 2 ω s 4 ω s Reducing ωs ( ωs < 2 ωo) moves repeated spectra too close: spectra interact (sum) Original signal cannot be reconstructed! Also note "amplified" foldover. 6 ω s

14 14 Aliasing Another view of prior example Frequencies Ω = Ω actual + k ωs, k = 0, ± 1, ±2,... possible aliases of Ω actual Aliasing example: sine waves Sampling at ωs = 2 π : can confuse sin( 2π 0.1 t) for sin( 2π 0.9 t) sin( 2π 0.1 t) alias of sin( 2π 0.9 t): both have same value at sample times

15 15 Aliasing example: Rotating wheel in motion picture Movie camera image data sampler: fs 30 frames per sec. reconstruct image on movie screen on screen: wheel of high speed vehicle rotates slowly or backwards Rotating wheel with spoke Different shades (colors) different sampling rates Color sampling frequency fs Appearance Comments (Green) (Blue) (Orange) (Red) (Yellow) 8 samples /rotation normal motion: correct frequency & direction 8 samples / (6 1/8) rotations retrograde (backwards) motion 4 x Nyquist obeys rule of thumb alias: f = f actual + k fs f actual = 1, fs = 1.3 f = - 1/8 1 sample / 1 rotation no motion alias: Ω = Ω actual + k ωs used to measure shaft speeds 8 samples / 9 rotations slow (forward) motion alias: Ω = Ω actual + k ωs 2 samples /1 rotation Correct frequency, cannot distinguish direction Limiting sampling rate at Nyquist freqency (theorem)

16 ALIASING & EFFECT OF DIFFERENT SAMPLING RATES

17 Signal Amplification & Anti-aliasing (pre)filters R(t) conditioning: anti-alias filtering & amplification r(t) avoid foldover of noise into signals attenuate high frequencies lowpass filter required: set ωc ωo Practical anti-aliasing pre-filter Types Butterworth Chebyschev Cauer Elliptic: set filter attenuation "floor" < LSB 2 (Least Significant Bit) Bessel Amplification: boost weak signal & tailor to A/D converter fill A/D converter range for maximum resolution dynamic range, amplified signal = full scale voltage range, converter (FS)

18 D/A Post processing c(t) conditioning & amplification C(t) Output of D/A converter with zero order hold: staircase steps may activate ringing (mechanical system natural frequencies) Prescription: passive 1st or 2nd order smoothing filter