Sampling and AliasingOutlineData ConversionSampling: Time DomainSampling: Frequency DomainSampling TheoremSlide 7Sampling and OversamplingAliasingSlide 10Slide 11Bandpass SamplingSampling for Up/DownconversionRolling Shutter CamerasRolling Shutter ArtifactsConclusionProf. Brian L. EvansDept. of Electrical and Computer EngineeringThe University of Texas at AustinLecture 4 http://www.ece.utexas.edu/~bevans/courses/realtimeEE 445S Real-Time Digital Signal Processing Lab Fall 2014Sampling and Aliasing4 - 2Outline•Data conversion•SamplingTime and frequency domainsSampling theorem•Aliasing•Bandpass sampling•Rolling shutter artifacts•ConclusionData Conversion•Analog-to-Digital ConversionLowpass filter hasstopband frequencyless than ½ fs to reducealiasing due to sampling(enforce sampling theorem) •Digital-to-Analog ConversionDiscrete-to-continuousconversion could be assimple as sample and holdLowpass filter has stopbandfrequency less than ½ fs reduce artificial high frequenciesAnalog Lowpass FilterDiscrete to Continuous ConversionfsLecture 7Analog Lowpass FilterQuantizerSampler at sampling rate of fsLecture 8Lecture 44 - 3Data Conversion4 - 4  sTnfnf Sampling: Time Domain•Many signals originate in continuous-timeTalking on cell phone, or playing acoustic music•By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbersn  {…, -2, -1, 0, 1, 2,…}Ts is the sampling period.Sampled analog waveform   nssampledTnttftf )(impulse trainf(t)tTsTs tfsampledSampling - Review4 - 5Sampling: Frequency Domain•Sampling replicates spectrum of continuous-time signal at integer multiples of sampling frequency•Fourier series of impulse train where s = 2  fs    ) (2cos2 ) (cos2 1 1 )( ...ttTTnttsssnsTs  ) (2cos)(2 ) (cos)(2 )( 1 )( )()( ... ttfttftfTttftgsssTsG()ssssF()2fmax-2fmaxmaxmaxmax2222 ifonly and if gap fffffssModulationby cos(2 s t)Modulationby cos(s t)How to recover F()?Sampling - Review4 - 6Sampling Theorem•Continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x(n Ts) if samples taken at rate fs > 2 fmaxNyquist rate = 2 fmaxNyquist frequency = fs / 2•Example: Sampling audio signalsNormal human hearing is from about 20 Hz to 20 kHzApply lowpass filter before sampling to pass low frequencies up to 20 kHz and reject high frequenciesLowpass filter needs 10% of maximum passband frequency to roll off to zero (2 kHz rolloff in this case)What happens if fs = 2 fmax?Sampling - Review4 - 7Sampling TheoremAssumption•Continuous-time signal has absolutely no frequency content above fmax•Sampling time is exactly the same between any two samples•Sequence of numbers obtained by sampling is represented in exact precision•Conversion of sequence to continuous time is idealIn PracticeSampling4 - 8Sampling and Oversampling•As sampling rate increases above Nyquist rate, sampled waveform looks more like original•Zero crossings: frequency content of a sinusoidDistance between two zero crossings: one half periodWith sampling theorem satisfied, sampled sinusoid crosses zero right number of times per periodIn some applications, frequency content matters not time-domain waveform shape•DSP First, Ch. 4, Sampling/Interpolation demoFor username/password help Samplinglinklink4 - 9Aliasing•Continuous-time sinusoidx(t) = A cos(2f0 t + )•Sample at Ts = 1/fsx[n] = x(Tsn) =A cos(2f0 Ts n + )•Keeping the sampling period same, sampley(t) = A cos(2(f0 + l fs) t + ) where l is an integery[n] = y(Tsn)= A cos(2(f0 + lfs)Tsn + )= A cos(2f0Tsn + 2lfsTsn + )= A cos(2f0Tsn + 2ln + )= A cos(2f0Tsn + )= x[n]Here, fsTs = 1Since l is an integer,cos(x + 2 l) = cos(x)• y[n] indistinguishable from x[n]Aliasing4 - 10Aliasing•Since l is any integer, a countable but infinite number of sinusoids give same sampled sequence•Frequencies f0 + l fs for l  0Called aliases of frequency f0 with respect to fsAll aliased frequencies appear same as f0 due to sampling•Signal Processing First, Continuous to Discrete Sampling demo (con2dis)Aliasinglink4 - 11Aliasing•Sinusoid sin(2  finput t) sampled at fs = 2000 samples/s with finput varied•Mirror image effect about finput = ½ fs gives rise to name of foldingApparentfrequency (Hz)Input frequency, finput (Hz)10001000 2000 3000 4000fs = 2000 samples/sAliasing4 - 12Bandpass Sampling•Reduce sampling rateBandwidth: f2 – f1Sampling rate fs mustbe greater than analogbandwidth fs > f2 – f1For replica to be centeredat origin after samplingfcenter = ½(f1 + f2) = k fs•Practical issuesSampling clock tolerance: fcenter = k fsEffects of noiseIdeal Bandpass Spectrumf1f2f–f2–f1Sample at fsSampled Ideal Bandpass Spectrumf1f2f–f2–f1Lowpass filter to extract basebandBandpass SamplingSampling for Up/Downconversion•Upconversion methodSampling plus bandpass filtering to extract intermediate frequency (IF) band with fIF = kIF fs•Downconversion methodBandpass sampling plus bandpass filtering to extract intermediate frequency (IF) band with fIF = kIF fsffmax-fmaxffsfIFfIFfsf1f2f–f2–f1Sample at fsf–f2–f1-fIFfIFBandpass SamplingRolling Shutter Cameras•Smart phone and point-and-shoot camerasNo (global) hardware shutter to reduce cost, size, weightLight continuously impinges on sensor arrayArtifacts due to relative motion between objects and cameraRolling Shutter ArtifactsFigure from tutorial by Forssen et al. at 2012 IEEE Conf. on Computer Vision & Pattern Recognition•Plucked guitar strings – global shutter cameraString vibration is (correctly) damped sinusoid vs. time•“Guitar Oscillations Captured with iPhone 4”Rolling shutter (sampling) artifacts but not aliasing effects•Fast camera motionPan camera fast left/rightPole wobbles and bendsBuilding skewedRolling Shutter ArtifactsRolling Shutter ArtifactsC. Jia and B. L. Evans, “Probabilistic 3-D Motion Estimation for Rolling Shutter Video Rectification from Visual and Inertial Measurements,” IEEE Multimedia Signal Proc. Workshop, 2012. Link to article. Warped frameCompensated using gyroscope readings (i.e. camera rotation) and video