Today Digital filters and signal processing Filter examples and properties FIR filters Filter design Implementation issuesDACsDACs PWMDSP Big PictureSignal Reconstruction Analog filter gets rid of unwanted high-frequency componentsData Acquisition Signal: Time-varying measurable quantity whose variation normally conveys information Quantity often a voltage obtained from some transducer  E.g. a microphone Analog signals have infinitely variable values at all timestimes Digital signals are discrete in time and in value Often obtained by sampling analog signals Sampling produces sequence of numbers• E.g. { ... , x[-2], x[-1], x[0], x[1], x[2], ... }  These are time domain signalsSampling Transducers Transducer turns a physical quantity into a voltage ADC turns voltage into an n-bit integer Sampling is typically performed periodically Sampling permits us to reconstruct signals from the world•E.g. sounds, seismic vibrations•E.g. sounds, seismic vibrations Key issue: aliasing Nyquist rate: 0.5 * sampling rate Frequencies higher than the Nyquist rate get mapped to frequencies below the Nyquist rate Aliasing cannot be undone by subsequent digital processingSampling Theorem Discovered by Claude Shannon in 1949:A signal can be reconstructed from its samples without loss of information, if the original signal has no frequencies above 1/2 the sampling frequency This is a pretty amazing result But note that it applies only to discrete time, not discrete valuesAliasing Details Let N be the sampling rate and F be a frequency found in the signal Frequencies between 0 and 0.5*N are sampled properly Frequencies >0.5*N are aliased• Frequencies between 0.5*N and N are mapped to (0.5*N)-F and have phase shifted 180°F and have phase shifted 180°• Frequencies between N and 1.5*N are mapped to f-N with no phase shift• Pattern repeats indefinitely Aliasing may or may not occur when N == F*2*X where X is a positive integerNo Aliasing1 kHz Signal, No AliasingAliasingMore AliasingN == 2*F ExampleAvoiding Aliasing1. Increase sampling rate Not a general-purpose solution• White noise is not band-limited• Faster sampling requires:– Faster ADC–Faster CPU–Faster CPU– More power– More RAM for buffering2. Filter out undesirable frequencies before sampling using analog filter(s) This is what is done in practice Analog filters are imperfect and require tradeoffsSignal Processing PragmaticsAliasing in Space Spatial sampling incurs aliasing problems also Example: CCD in digital camera samples an image in a grid pattern Real world is not band-limited Can mitigate aliasing by increasing sampling rateSamples PixelPoint vs. SupersamplingPoint sampling 4x4 SupersamplingDigital Signal Processing Basic idea  Digital signals can be manipulated losslessly SW control gives great flexibility DSP examples Amplification or attenuationFiltering –leaving out some unwanted part of the signalFiltering –leaving out some unwanted part of the signal Rectification – making waveform purely positive Modulation – multiplying signal by another signal• E.g. a high-frequency sine waveAssumptions1. Signal sampled at fixed and known rate fs I.e., ADC driven by timer interrupts2. Aliasing has not occurred I.e., signal has no significant frequency components greater than 0.5*fgreater than 0.5*fs These have to be removed before ADC using an analog filter Non-significant signals have amplitude smaller than the ADC resolutionFilter Terms for CS People Low pass – lets low frequency signals through, suppresses high frequency High pass – lets high frequency signals through, suppresses low frequency Passband – range of frequencies passed by a filter Stopband – range of frequencies blocked Transition band – in between theseSimple Digital Filters y(n) = 0.5 * (x(n) + x(n-1)) Why not use x(n+1)? y(n) = (1.0/6) * (x(n) + x(n-1) + x(n-2) + … + y(n-5) ) y(n) = 0.5 * (x(n) + x(n-3))y(n) = 0.5 * (y(n-1) + x(n))y(n) = 0.5 * (y(n-1) + x(n)) What makes this one different? y(n) = median [ x(n) + x(n-1) + x(n-2) ]Gain vs. FrequencyGain1.01.5y(n) =(y(n-1)+x(n))/2y(n) =(x(n)+x(n-1))/2y(n) =(x(n)+x(n-1)+x(n-2)+ x(n-3)+x(n-4)+x(n-5))/6Gain0.00.50.0 0.1 0.2 0.3 0.4 0.5frequency f/fsy(n) =(x(n)+x(n-3))/2Useful Signals Step:  …, 0, 0, 0, 1, 1, 1, …1Step s(n) Impulse:  …, 0, 0, 0, 1, 0, 0, …-3 -2 -1 0 1 2 31Impulse i(n)-10 -2-31 2 3Step Response0.60.81step inputFIRResponse0 1 2 3 4 500.20.4sample number, nIIRmedianResponseImpulse Response0.60.81impulse inputFIRResponse0 1 2 3 4 500.20.4sample number, nFIRIIRmedianResponseFIR Filters Finite impulse response Filter “remembers” the arrival of an impulse for a finite time Designing the coefficients can be hard Moving average filter is a simple example of FIRMoving Average ExampleFIR in CSAMPLE fir_basic (SAMPLE input, int ntaps, const SAMPLE coeff[], SAMPLE z[]) {z[0] = input;SAMPLE accum = 0; SAMPLE accum = 0; for (int ii = 0; ii < ntaps; ii++) { accum += coeff[ii] * z[ii]; }for (ii = ntaps - 2; ii >= 0; ii--) { z[ii + 1] = z[ii]; } return accum; }Implementation Issues Usually done with fixed-point How to deal with overflow? A few optimizations Put coefficients in registers Put sample buffer in registers Block filter• Put both samples and coefficients in registers• Unroll loops Hardware-supported circular buffers  Creating very fast FIR implementations is importantFilter Design Where do coefficients come from for the moving average filter? In general:1. Design filter by hand2. Use a filter design toolFew filters designed by hand in practiceFew filters designed by hand in practice Filters design requires tradeoffs between1. Filter order2. Transition width3. Peak ripple amplitude Tradeoffs are inherentFilter Design in Matlab Matlab has excellent filter design support C = firpm (N, F, A) N = length of filter - 1 F = vector of frequency bands normalized to Nyquist A = vector of desired amplitudesfirpmuses minimax –it minimizes the maximum firpmuses minimax –it minimizes the maximum deviation from the desired amplitudeFilter Design Examplesf = [ 0.0 0.3 0.4 0.6 0.7 1.0]; a = [ 0 0 1 1 0 0];fil1 = firpm( 10, f, a);fil2 =