http://elec3004.org Sampling & More 2014 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 1 2-Mar Introduction 3-Mar Systems Overview 2 9-Mar Signals as Vectors & Systems as Maps 10-Mar [Signals] 3 4 5 6 7 8 9 10 11 12 13 Sampling & Data Acquisition & 16-Mar Antialiasing Filters 17-Mar [Sampling] 23-Mar System Analysis & Convolution 24-Mar [Convolution & FT] 30-Mar Frequency Response & Filter Analysis 31-Mar [Filters] 13-Apr Discrete Systems & Z-Transforms 14-Apr [Z-Transforms] 20-Apr Introduction to Digital Control 21-Apr [Feedback] 27-Apr Digital Filters 28-Apr [Digital Filters] 4-May Digital Control Design 5-May [Digitial Control] 11-May Stability of Digital Systems 12-May [Stability] 18-May State-Space 19-May Controllability & Observability 25-May PID Control & System Identification 26-May Digitial Control System Hardware 31-May Applications in Industry & Information Theory & Communications 2-Jun Summary and Course Review ELEC 3004: Systems 16 March 2015-2 1
Interpretations of Systems as Maps ELEC 3004: Systems 16 March 2015-3 Then a System is a MATRIX ELEC 3004: Systems 16 March 2015-4 2
System Analysis [Chapter 2, Lathi] ELEC 3004: Systems 16 March 2015-5 Linear Differential Systems ELEC 3004: Systems 16 March 2015-6 3
Linear Differential System Order y(t)=p(d)/q(d) f(t) P(D): M Q(D): N In practice: m n (yes, N is denominator) if m > n: then the system is an (m - n) th -order differentiator of high-frequency signals! Derivatives magnify noise! ELEC 3004: Systems 16 March 2015-7 Derivatives magnify noise! 1 sin(10 t) + 0.1 sin(100 t) 10 10 cos(10 t) 0.5 5 0 0-0.5-5 1-1 -6-4 -2 0 t t) + 2 sin(100 4 6 sin(10 0.1 t) 20 15-10 -6-4 -2 0 2 4 6 10 cos(10 t t) + 10 cos(100 t) 0.5 10 5 0 0-5 -0.5-10 -1-15 -6-4 -2 0 2 4 6 t -20-6 -4-2 0 2 4 6 t ELEC 3004: Systems 16 March 2015-8 4
Zero-Input Zero-State Zero Input = The system response when the input f(t) = 0 so that it is the result of internal system conditions (such as energy storages, initial conditions) alone. It is independent of the external input. Zero-State = the system response to the external input f (t) when the system is in zero state, meaning the absence of all internal energy storages; that is, all initial conditions are zero. ELEC 3004: Systems 16 March 2015-9 System Stability Lathi, p. 149 ELEC 3004: Systems 16 March 2015-10 5
System Stability [II] Lathi, p. 150 ELEC 3004: Systems 16 March 2015-11 System Stability [III] ELEC 3004: Systems 16 March 2015-12 6
Signals Review ELEC 3004: Systems 16 March 2015-13 Signal: A carrier of (desired) information [1] Need NOT be electrical: Thermometer Clock hands Automobile speedometer Need NOT always being given Abnormal sounds/operations Ex: pitch or engine hum during machining as an indicator for feeds and speeds ELEC 3004: Systems 16 March 2015-14 7
Signal: A carrier of (desired) information [2] Electrical signals Voltage Current Digital signals Convert analog electrical signals to an appropriate digital electrical message Processing by a microcontroller or microprocessor ELEC 3004: Systems 16 March 2015-15 Ex: Current-to-voltage conversion simple: Precision Resistor better: Use an op amp ELEC 3004: Systems 16 March 2015-16 8
Sampling! ELEC 3004: Systems 16 March 2015-30 Not this type of sampling ELEC 3004: Systems 16 March 2015-31 9
This type of sampling Source: Wikipedia: http://en.wikipedia.org/wiki/file:signal_sampling.png ELEC 3004: Systems 16 March 2015-32 Analog vs Digital Analog Signal: An analog or analogue signal is any variable signal continuous in both time and amplitude Digital Signal: A digital signal is a signal that is both discrete and quantized E.g. Music stored in a CD: 44,100 Samples per second and 16 bits to represent amplitude ELEC 3004: Systems 16 March 2015-33 10
Expected signal (mv) Expected signal (mv) Digital Signal Representation of a signal against a discrete set The set is fixed in by computing hardware Can be scaled or normalized but is limited Time is also discretized ELEC 3004: Systems 16 March 2015-34 Representation of Signal Time Discretization Digitization 600 Coarse time discretization 600 Coarse signal digitization 500 500 400 400 300 300 200 200 100 True signal Discrete time sampled points 0 0 5 10 15 time (s) 100 True signal Digitization 0 0 5 10 15 time (s) ELEC 3004: Systems 16 March 2015-35 11
Mathematics of Sampling and Reconstruction sampling reconstruction x(t) x c (t) DSP Ideal y(t) LPF Impulse train T (t)= (t - n t) t Sampling frequency f s = 1/ t 1 0 Gain f c Freq Cut-off frequency = f c ELEC 3004: Systems 16 March 2015-36 Mathematical Model of Sampling x(t) multiplied by impulse train T(t) x ( t) c x( t) ( t) ( t t) ( t 2 t) x( n t) ( t n t) x c (t) is a train of impulses of height x(t) t=n t n x( t) ( t) T ELEC 3004: Systems 16 March 2015-37 12
Amplitude x c (t) x(t) 2 Continuous-time 1 0-1 -2-10 -8-6 -4-2 0 2 4 6 8 10 t Discrete-time 2 1 0-1 -2-10 -8-6 -4-2 0 2 4 6 8 10 t ELEC 3004: Systems 16 March 2015-38 Discrete Time Signal Image a signal 1 Signal Digitized Signal 0.5 0-0.5-1 -8-6 -4-2 0 2 4 6 8 time (s) ELEC 3004: Systems 16 March 2015-39 13
Amplitude Amplitude Discrete Time Signals Digitization helps beat the Noise! 1.5 1 Signal + 5% Gausian Noise Digitized Noisy Signal 0.5 0-0.5-1 -1.5-8 -6-4 -2 0 2 4 6 8 time (s) ELEC 3004: Systems 16 March 2015-40 Discrete Time Signals But only so much 1.5 1 Signal + 20% Gausian Noise Digitized Noisy Signal 0.5 0-0.5-1 -1.5-8 -6-4 -2 0 2 4 6 8 time (s) ELEC 3004: Systems 16 March 2015-41 14
Discrete Time Signals Can make control tricky! ELEC 3004: Systems 16 March 2015-42 Signal Manipulations Shifting Reversal Time Scaling (Down Sampling) (Up Sampling) ELEC 3004: Systems 16 March 2015-43 15
Frequency Domain Analysis of Sampling Consider the case where the DSP performs no filtering operations i.e., only passes xc(t) to the reconstruction filter To understand we need to look at the frequency domain Sampling: we know multiplication in time convolution in frequency F{x(t)} = X(w) F{ T(t)} = (w - 2 n/ t), i.e., an impulse train in the frequency domain ELEC 3004: Systems 16 March 2015-44 Frequency Domain Analysis of Sampling In the frequency domain we have 1 2 2 n X c( w) X ( w)* w 2 t n t 1 2 n X w t n t Remember convolution with an impulse? Same idea for an impulse train Let s look at an example where X(w) is triangular function with maximum frequency w m rad/s being sampled by an impulse train, of frequency w s rad/s ELEC 3004: Systems 16 March 2015-45 16
Fourier transform of original signal X(ω) (signal spectrum) Fourier transform of impulse train T ( /2 ) (sampling signal) 0 w s = 2 / t 4 / t Fourier transform of sampled signal 1/ t Original Replica 1 Replica 2 w w Original spectrum convolved with spectrum of impulse train ELEC 3004: Systems 16 March 2015-46 Spectrum of sampled signal 1/ t Original Replica 1 Replica 2 Reconstruction filter (ideal lowpass filter) w t -w c w c = w m Spectrum of reconstructed signal w Reconstruction filter removes the replica spectrums & leaves only the original -w m w m ELEC 3004: Systems 16 March 2015-47 w 17
Sampling Frequency In this example it was possible to recover the original signal from the discrete-time samples But is this always the case? Consider an example where the sampling frequency w s is reduced i.e., t is increased ELEC 3004: Systems 16 March 2015-48 Original Spectrum -w m w m w Fourier transform of impulse train (sampling signal) 0 2 / t 4 / t 6 / t Amplitude spectrum of sampled signal w Replica spectrums overlap with original (and each other) This is Aliasing w Original Replica 1 Replica 2 16 March 2015 - ELEC 3004: Systems 49 18
Amplitude spectrum of sampled signal Original Replica 1 Replica 2 Reconstruction filter (ideal lowpass filter) sampled signal spectrum w -w c w c = w m Spectrum of reconstructed signal The effect of aliasing is that higher frequencies of alias to (appear as) lower frequencies -w m w m w Due to overlapping replicas (aliasing) the reconstruction filter cannot recover the original spectrum ELEC 3004: Systems 16 March 2015-50 w Sampling Theorem The Nyquist criterion states: To prevent aliasing, a bandlimited signal of bandwidth w B rad/s must be sampled at a rate greater than 2w B rad/s w s > 2w B Note: this is a > sign not a Also note: Most real world signals require band-limiting with a lowpass (anti-aliasing) filter ELEC 3004: Systems 16 March 2015-51 19
Time Domain Analysis of Sampling Frequency domain analysis of sampling is very useful to understand sampling (X(w)* (w - 2 n/ t) ) reconstruction (lowpass filter removes replicas) aliasing (if w s 2w B ) Time domain analysis can also illustrate the concepts sampling a sinewave of increasing frequency sampling images of a rotating wheel ELEC 3004: Systems 16 March 2015-52 Original signal Discrete-time samples Reconstructed signal A signal of the original frequency is reconstructed ELEC 3004: Systems 16 March 2015-53 20
signal Original signal Discrete-time samples Reconstructed signal A signal with a reduced frequency is recovered, i.e., the signal is aliased to a lower frequency (we recover a replica) ELEC 3004: Systems 16 March 2015-54 Sampling < Nyquist Aliasing 1.5 1 0.5 0-0.5-1 True signal Aliased (under sampled) signal -1.5 0 5 10 15 time ELEC 3004: Systems 16 March 2015-55 21
Normalized magnitude Normalized magnitude Nyquist is not enough 1 1Hz Sin Wave: Sin(2 t) 2 Hz Sampling 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 1 2 3 4 5 6 7 Time(s) ELEC 3004: Systems 16 March 2015-56 A little more than Nyquist is not enough 1 1Hz Sin Wave: Sin(2 t) 4 Hz Sampling 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 1 2 3 4 5 6 7 Time(s) ELEC 3004: Systems 16 March 2015-57 22
Sampled Spectrum w s > 2wm LPF -w m w m w s orignal replica 1 original freq recovered Sampled Spectrum w s < 2w m LPF -w m w m w s w Original and replica spectrums overlap Lower frequency recovered (w s w m ) w orignal replica 1 ELEC 3004: Systems 16 March 2015-58 Temporal Aliasing 90 o clockwise rotation/frame clockwise rotation perceived 270 o clockwise rotation/frame (90 o ) anticlockwise rotation perceived i.e., aliasing Require LPF to blur motion ELEC 3004: Systems 16 March 2015-59 23
Time Domain Analysis of Reconstruction Frequency domain: multiply by ideal LPF ideal LPF: rect function (gain t, cut off w c ) removes replica spectrums, leaves original Time domain: this is equivalent to convolution with sinc function as F -1 { t rect(w/w c )} = t w c sinc(w c t/ ) i.e., weighted sinc on every sample Normally, w c = w s /2 x ( t) r n x( n t) tw c w sinc c ( t n t) ELEC 3004: Systems 16 March 2015-60 Reconstruction ELEC 3004: Systems 16 March 2015-61 24
Reconstruction Zero-Order Hold [ZOH] ELEC 3004: Systems 16 March 2015-62 Reconstruction Whittaker Shannon interpolation formula ELEC 3004: Systems 16 March 2015-63 25
Value Reconstruction Whittaker Shannon interpolation formula ELEC 3004: Systems 16 March 2015-64 Ideal "sinc" Interpolation of sample values [0 0 0.75 1 0.5 0 0] 1 0.8 reconstructed signal x r (t) 0.6 0.4 0.2 0-0.2-4 -3-2 -1 0 1 2 3 4 Sample ELEC 3004: Systems 16 March 2015-65 26
Amplitude (V) Sampling and Reconstruction Theory and Practice Signal is bandlimited to bandwidth WB Problem: real signals are not bandlimited Therefore, require (non-ideal) anti-aliasing filter Signal multiplied by ideal impulse train problems: sample pulses have finite width and not in practice, but sample & hold circuit Samples discrete-time, continuous valued Problem: require discrete values for DSP Therefore, require A/D converter (quantisation) Ideal lowpass reconstruction ( sinc interpolation) problems: ideal lowpass filter not available Therefore, use D/A converter and practical lowpass filter ELEC 3004: Systems 16 March 2015-66 16 14 staircase output from D/A converter (ZOH) output samples D/A output 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Time (sec) ELEC 3004: Systems 16 March 2015-67 27
Amplitude (V) Amplitude (V) 16 Smooth output from reconstruction filter D/A output Reconstruction filter output 14 12 10 8 6 4 2 0 2 4 6 8 10 12 Time (sec) ELEC 3004: Systems 16 March 2015-68 16 Example: error due to signal quantisation original signal x(t) quantised samples x q (t) 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Sample number ELEC 3004: Systems 16 March 2015-69 28
Original Signal After Anti-aliasing LPF After Sample & Hold After Reconstruction LPF After D/A After A/D Complete practical DSP system signals DSP ELEC 3004: Systems 16 March 2015-70 Zero Order Hold (ZOH) ZOH impulse response ZOH amplitude response ZOH phase response ELEC 3004: Systems 16 March 2015-71 29
Finite Width Sampling Impulse train sampling not realisable sample pulses have finite width (say nanosecs) This produces two effects, Impulse train has sinc envelope in frequency domain impulse train is square wave with small duty cycle Reduces amplitude of replica spectrums smaller replicas to remove with reconstruction filter Averaging of signal during sample time effective low pass filter of original signal can reduce aliasing, but can reduce fidelity negligible with most S/H ELEC 3004: Systems 16 March 2015-72 Aliasing: Another view of this ELEC 3004: Systems 16 March 2015-73 30
Alliasing Aliasing - through sampling, two entirely different analog sinusoids take on the same discrete time identity For f[k]=cosωk, Ω=ωT: The period has to be less than Fh (highest frequency): Thus: ω f : aliased frequency: ELEC 3004: Systems 16 March 2015-74 Practical Anti-aliasing Filter Non-ideal filter wc = ws /2 Filter usually 4th 6th order (e.g., Butterworth) so frequencies > wc may still be present not higher order as phase response gets worse Luckily, most real signals are lowpass in nature signal power reduces with increasing frequency e.g., speech naturally bandlimited (say < 8KHz) Natural signals have a (approx) 1/f spectrum so, in practice aliasing is not (usually) a problem ELEC 3004: Systems 16 March 2015-75 31
Amplitude spectrum of original signal -w m w m w Fourier transform of sampling signal (pulses have finite width) 0 w s = 2 / t 4 / t Fourier transform of sampled signal 1/ t sinc envelope Zero at harmo 1/duty cycle w Original Replica 1 Replica 2 ELEC 3004: Systems 16 March 2015-76 w Practical Sampling Sample and Hold (S/H) 1. takes a sample every t seconds 2. holds that value constant until next sample Produces staircase waveform, x(n t) sample instant x(n t) x(t) hold for t t ELEC 3004: Systems 16 March 2015-77 32
Quantisation Analogue to digital converter (A/D) Calculates nearest binary number to x(n t) x q [n] = q(x(n t)), where q() is non-linear rounding fctn output modeled as x q [n] = x(n t) + e[n] Approximation process therefore, loss of information (unrecoverable) known as quantisation noise (e[n]) error reduced as number of bits in A/D increased i.e., x, quantisation step size reduces e[ n] x 2 ELEC 3004: Systems 16 March 2015-78 Input-output for 4-bit quantiser (two s compliment) 2A x m 2 1 where A = max amplitude m = no. quantisation bits Digital 7 0111 6 0110 5 0101 4 0100 3 0011 2 0010 1 0001 0 0000-1 1111-2 1110-3 1101-4 1100-5 1011-6 1010-7 1000 x quantisation step size Analogue ELEC 3004: Systems 16 March 2015-79 33
Signal to Quantisation Noise To estimate SQNR we assume e[n] is uncorrelated to signal and is a uniform random process assumptions not always correct! not the only assumptions we could make Also known a Dynamic range (R D ) expressed in decibels (db) ratio of power of largest signal to smallest (noise) P R D 10log10 P signal noise ELEC 3004: Systems 16 March 2015-80 Dynamic Range Need to estimate: 1. Noise power uniform random process: P noise = x 2 /12 2. Signal power (at least) two possible assumptions 1. sinusoidal: P signal = A 2 /2 2. zero mean Gaussian process: P signal = 2 Note: as A/3: P signal A 2 /9 where 2 = variance, A = signal amplitude 1 extra bit halves x i.e., 20log10(1/2) = 6dB Regardless of assumptions: R D increases by 6dB for every bit that is added to the quantiser ELEC 3004: Systems 16 March 2015-81 34
Practical Reconstruction Two stage process: 1. Digital to analogue converter (D/A) zero order hold filter produces staircase analogue output 2. Reconstruction filter non-ideal filter: w c = w s /2 further reduces replica spectrums usually 4 th 6 th order e.g., Butterworth for acceptable phase response ELEC 3004: Systems 16 March 2015-82 D/A Converter Analogue output y(t) is convolution of output samples y(n t) with h ZOH (t) y( t) y( n t) h h H ZOH ZOH n ZOH ( t n t) 1, 0 t t ( t) 0, otherwise jw t sin( w t / 2) ( w) t exp 2 w t / 2 D/A is lowpass filter with sinc type frequency response It does not completely remove the replica spectrums Therefore, additional reconstruction filter required ELEC 3004: Systems 16 March 2015-83 35
Summary Theoretical model of Sampling bandlimited signal (wb) multiplication by ideal impulse train (ws > 2wB) convolution of frequency spectrums (creates replicas) Ideal lowpass filter to remove replica spectrums wc = ws /2 Sinc interpolation Practical systems Anti-aliasing filter (wc < ws /2) A/D (S/H and quantisation) D/A (ZOH) Reconstruction filter (wc = ws /2) Don t confuse theory and practice! ELEC 3004: Systems 16 March 2015-84 36