Spectrograms Overview Introduction and motivation Windowed estimates as filter banks Uncertainty principle Examples Other estimates

Transcription

1 Spectrograms Overview Introduction and motivation Windowed estimates as filter banks Uncertainty principle Examples Other estimates J. McNames Portland State University ECE 3 Spectrograms Ver

2 Introduction and Motivation In practical applications, many the frequency content of signals changes over time These signals are often described as nonstationary In order to apply the techniques, we usually assume the signal statistics change slowly Locally stationary Rate of change depends on the process Spectrograms estimate how these properties change with time J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

3 x[n] = 1 π Practical Application X(e jω ) e jωn dω X(e jω ) = π + n= x[n] e jωn In practice, time-frequency analysis is most frequently performed by displaying the squared magnitude of segments of the signal Often called sliding windows Resulting DTFT s are a function of two independent variables: time and frequency X(e jω, n) = n+n/ k=n N/ w[k]x[k] e jωk where w[k] is a user-specified window function J. McNames Portland State University ECE 3 Spectrograms Ver

4 Comments on Spectrograms X(e jω, n) = n+n/ k=n N/ w[k]x[k] e jωk The spectrogram only shows the magnitude X(e jω, n) or the squared magnitude X(e jω, n) The squared magnitude can be interpreted as an estimate of the power spectral density (PSD) The phase is never plotted No clear means of interpretation Typically represented with three-dimensional meshes, contour plots, or pseudo-colored images When used to estimate the time-varying PSD, are called spectrograms or short-time Fourier transforms (STFT) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

5 Example 1: Chirp Spectrogram Apply the windowed DTFT to a chirp signal with a linear frequency sweep from 1 1 Hz over a period of s. J. McNames Portland State University ECE 3 Spectrograms Ver

6 Example 1: Chirp Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

7 Example 1: Chirp Spectrogram T w =.5 s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

8 Example 1: Chirp Spectrogram T w = 1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

9 Example 1: Chirp Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

10 Example 1: Chirp Spectrogram T w = 5. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

11 Example 1: Chirp Spectrogram T w =1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

12 Example 1: Chirp Spectrogram T w =15. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

13 J. McNames Portland State University ECE 3 Spectrograms Ver

14 Example 1: Relevant MATLAB Code clear; close all; %============================================================================== % User- Specified Parameters %============================================================================== fs = ; % Sample rate (Hz) T = ; % Signal duration (s) f = [1 1]; % Chirp frequencies [start stop] Tw = [ ]; % Window lengths k = 1:T*fs; t = (k-.5)/fs; % Discrete time index % Sample times (s) %============================================================================== % Preprocessing %============================================================================== functionname = sprintf( %s,mfilename); % Get the function name %============================================================================== % Main Loop %============================================================================== for c=1: % DEBUG switch c case 1, x = chirp(t,f(1),t,f()); sg = Chirp ; J. McNames Portland State University ECE 3 Spectrograms Ver

15 end case, x = cos(*pi*5*t); sg = Cosine ; case 3, [jnk,id] = min(abs(t-t/)); x = zeros(length(k),1); x(id) = 1; sg = Impulse ; case, id = find(t>t/); x = zeros(length(k),1); x(id) = 1; sg = UnitStep ; case 5, id = 1:1*fs:max(k); x = zeros(length(k),1); x(id) = 1; sg = ImpulseTrain ; case, x = randn(length(k),1); sg = WhiteNoise ; fileidentifier = fopen([functionname - sg.tex ], w ); for c1=1:length(tw), tw = Tw(c1); [S,ts,f] = NonparametricSpectrogram(x,fs,tw); figure; FigureSet(1, Slides ); J. McNames Portland State University ECE 3 Spectrograms Ver

16 I = abs(s).^; cr = prctile(reshape(i,prod(size(i)),1),[ ]); ha = axes( Position,[ ]); h = imagesc(ts,f,abs(s).^,cr); hold on; h = plot(tw/*[1 1],[ fs/], k,tw/*[1 1],[ fs/], w ); set(h(1), LineWidth,); set(h(), LineWidth,1); h = plot((t-tw/)*[1 1],[ fs/], k,(t-tw/)*[1 1],[ fs/], w ); set(h(1), LineWidth,); set(h(), LineWidth,1); hold off; set(gca, YDir, Normal ); set(gca, XAxisLocation, Top ); set(gca, YAxisLocation, Right ); xlim([ T]); ylim([ fs/]); %ht = title(sprintf( %s Signal T_w=%.1f s,sg,tw)); ha = axes( Position,[ ]); h = plot((k-.5)/fs,x, b ); set(h, LineWidth,.5); xlim([ T]); set(ha, YTick,[]) xlabel( Time (s) ); ha = axes( Position,[ ]); Rx = mean(i,); h = plot(rx,f, r ); set(h, LineWidth,.5); ylim([ fs/]); J. McNames Portland State University ECE 3 Spectrograms Ver

17 set(ha, XTick,[]) ylabel( ); AxisSet(); %set(ht, VerticalAlignment, Top ) end filename = sprintf( %s-%s-window%d,functionname,sg,c1); print(filename, -depsc ); fprintf(fileidentifier, %%========================================================================= fprintf(fileidentifier, \\newslide\n ); fprintf(fileidentifier, \\slideheading{example \\arabic{exc}: %s Spectrogram $T_w$=%.1f s}\n,sg,t fprintf(fileidentifier, %%========================================================================= fprintf(fileidentifier, \\includegraphics[scale=1]{matlab/%s}\n,filename); %print(sprintf( %sex%3d,sg,tw*1), -depsc ); end fclose(fileidentifier); J. McNames Portland State University ECE 3 Spectrograms Ver

18 Why the Windowed Periodogram? Why window the segments? Could use a rectangular window If tapered window is not applied, estimate will vary even if the signal is a constant sinusoid Due to variation of phase of sinusoid that is included in the segment (covered by the window) Taper reduces this effect Also reduces sidelobe leakage (more later) J. McNames Portland State University ECE 3 Spectrograms Ver

19 Filter Bank Interpretation X(e jω, n) = x(n + l)w(l)e jωl = = l= m= m= x(m)w(m n)e jω(m n) x(m)w( (n m))e jω(n m) = x(n) w( n)e jωn The spectrogram can be thought of as the squared magnitude of the output of an LTI system with impulse response h Ω (n) = w( n)e jωn The magnitude response evaluated at Ω o is then W ( e j(ω o Ω) ) J. McNames Portland State University ECE 3 Spectrograms Ver

20 Filter Bank Frequency Responses without Padding FFT Filters Blackman Window L=17 P= W(e j ω ) Frequency (radians/sample) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

21 Effect of Zero Padding.5 Time Domain 3 Frequency Domain 1 points points (11 zeros) Frequency Domain J. McNames Portland State University ECE 3 Spectrograms Ver

22 Filter Bank Frequency Responses with Padding FFT Filters Blackman Window L=17 P=3 1 1 W(e j ω ) Frequency (radians/sample) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

23 Primary Tradeoff: Segment Length Primary purpose of windowing is to guarantee local stationarity Window shape controls tradeoff between sidelobe leakage and main lob width Primary parameter is the window length Long window Poor time resolution Good frequency resolution Use with slowly varying signal characteristics Short window Good time resolution Poor frequency resolution Use with rapidly changing signal characteristics Thus L primarily controls the tradeoff between time and frequency resolution J. McNames Portland State University ECE 3 Spectrograms Ver

24 Uncertainty Principle Let σ t,x represent the duration of a signal and σ Ω,x represent the bandwidth of a signal. It can be shown that σ t,x σ Ω,x 1 In this context, it is not actually an expression of uncertainty Signals with short duration have broad bandwidth Signals with long duration can have narrow bandwidth Equivalently, if one density is narrow then the other is wide They can t both be arbitrarily narrow In quantum mechanics, applies to probability densities of position and velocity so uncertainty is appropriate J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

25 Time Bandwidth Product L=33 Windowed Sinusoid ω =. rad/sample X(e jω 1 5 Frequency Domain 1 3 L=5 L= Sample Index (n) X(e jω X(e jω Frequency (rad/sec) J. McNames Portland State University ECE 3 Spectrograms Ver

26 MATLAB Code %function [] = TimeBandwidthEx(); close all; NZ = ^1; w = pi/5; L = [3 1]+1; nl = length(l); k = (:NZ-1) ; w = k**pi/nz; kw = 1:NZ/; l = 11; wn = blackman(l); wn = wn*sqrt(l/sum(wn.^)); n = (-(l-1)/:(l-1)/) ; x = cos(w*n).*wn; ymx = max(x); ymn = min(x); yrg = ymx-ymn; figure; FigureSet(1, LTX ); for c1=1:nl, l = L(c1); n = (-(l-1)/:(l-1)/) ; J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

27 wn = blackman(l+); wn = wn(:end-1); % Trim off the zeros wn = wn*sqrt(l/sum(wn.^)); % Scale appropriately x = cos(w*n).*wn; % Windowed signal segment X = exp(j*(l-1)/*w).*fft(x,nz); % Calculate frequency response %disp(max(imag(x))); X = real(x); % Eliminate imaginary part due to finite resolution subplot(nl,,c1*-1); h = plot(n,x); xlim([-max(l) max(l)]); ylim([ymn-.5*yrg ymx+.5*yrg]); AxisLines; box off; ylabel(sprintf( L=%d,l)); if c1==1, title(sprintf( Windowed Sinusoid \\omega_=%5.3f rad/sample,w)); elseif c1==nl, xlabel( Sample Index (n) ); end; subplot(nl,,c1*); h = plot(w(kw),x(kw), r ); xlim([ pi]); ylim([ 1.5*max(X)]); AxisLines; ylabel( X(e^{j\omega} ); box off; if c1==1, title( Frequency Domain ); elseif c1==nl, xlabel( Frequency (rad/sec) ); end; J. McNames Portland State University ECE 3 Spectrograms Ver

28 end; AxisSet(); print( TimeBandwidthEx, -depsc ); J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

29 Attaining the Uncertainty Principle The Gaussian function attains the uncertainty principle with equality x(t) = 1 t e σ πσ X(jΩ) = e Ω σ e x (t) = 1 πσ e t σ E x (Ω) = σ t,x = σ σ Ω,x = 1 σ 1 πσ 1 e Ω σ σ t,x σ Ω,x = 1 If this has the best duration-bandwidth product (i.e., resolution), why don t we use this as a window? J. McNames Portland State University ECE 3 Spectrograms Ver

30 Practical Implementation of Spectrograms Remove the signal mean (center the signal) Apply a highpass filter, if necessary The MATLAB implementation is not very good Time and frequency resolution should not exceed that of the display Use sufficient zero padding and time resolution Make user-specified parameters Do not ask for degree of overlap - calculate it Edge effects Notify user of the extent of edge effects Handle edges elegantly (repeat last value, extrapolate by prediction, etc.) J. McNames Portland State University ECE 3 Spectrograms Ver

31 Example : Cosine, Impulse, and Step Spectrograms Repeat the previous example for a 5 Hz sinusoidal signal, the unit impulse at t = 1 s, and the unit step starting at t = 1 s. J. McNames Portland State University ECE 3 Spectrograms Ver

32 Example : Cosine Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

33 Example : Cosine Spectrogram T w =.5 s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

34 Example : Cosine Spectrogram T w = 1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

35 Example : Cosine Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

36 Example : Cosine Spectrogram T w = 5. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

37 Example : Cosine Spectrogram T w =1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

38 Example : Cosine Spectrogram T w =15. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

39 J. McNames Portland State University ECE 3 Spectrograms Ver

40 Example : Impulse Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

41 Example : Impulse Spectrogram T w =.5 s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

42 Example : Impulse Spectrogram T w = 1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

43 Example : Impulse Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

44 Example : Impulse Spectrogram T w = 5. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

45 Example : Impulse Spectrogram T w =1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

46 Example : Impulse Spectrogram T w =15. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

47 J. McNames Portland State University ECE 3 Spectrograms Ver

48 Periodic Signals Periodic signals consist of frequency components only at multiples of the fundamental frequency If the window length is smaller than the fundamental period, these areas analyzed as discrete events (insufficient frequency resolution) If the window length is several times the fundamental period, these areas appear to be periodic J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

49 Example 3: Impulse Train Spectrograms Repeat the previous example for a 1 Hz impulse train. J. McNames Portland State University ECE 3 Spectrograms Ver

50 Example 3: ImpulseTrain Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

51 Example 3: ImpulseTrain Spectrogram T w =.5 s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

52 Example 3: ImpulseTrain Spectrogram T w = 1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

53 Example 3: ImpulseTrain Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

54 Example 3: ImpulseTrain Spectrogram T w = 5. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

55 Example 3: ImpulseTrain Spectrogram T w =1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

56 Example 3: ImpulseTrain Spectrogram T w =15. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

57 J. McNames Portland State University ECE 3 Spectrograms Ver

58 Random Signals It s important to be able to distinguish significant features from random fluctuations due to estimation error White noise is a signal with random sample values Often, but not necessarily, from a Gaussian distribution Helpful to examine the spectrograms of white noise J. McNames Portland State University ECE 3 Spectrograms Ver

59 Example : White Noise Spectrograms Repeat the previous example for a white noise signal. J. McNames Portland State University ECE 3 Spectrograms Ver

60 Example : WhiteNoise Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

61 Example : WhiteNoise Spectrogram T w =.5 s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

62 Example : WhiteNoise Spectrogram T w = 1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

63 Example : WhiteNoise Spectrogram T w =. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

64 Example : WhiteNoise Spectrogram T w = 5. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

65 Example : WhiteNoise Spectrogram T w =1. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

66 Example : WhiteNoise Spectrogram T w =15. s Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

67 J. McNames Portland State University ECE 3 Spectrograms Ver

68 Example 5: Detecting Sleep Apnea Can obstructive sleep apnea be detected from the electrocardiogram alone? If possible, could build simple and cheap, portable diagnostic device International competition in Data set 5 recordings labelled by expert 1 unlabelled recording Approximately hours each Objective: detect OSA for every minute of the 1 recordings J. McNames Portland State University ECE 3 Spectrograms Ver. 1.1

69 Example 5: Detecting Sleep Apnea Our approach was based on spectrograms Implemented QRS detector Applied spectrogram to instantaneous heart rate and other features Interbeat intervals (R-R) S amplitude Pulse energy Winning entry! 9.% of minutes classified correctly Third attempt J. McNames Portland State University ECE 3 Spectrograms Ver

70 Example 5: Apnea Detection Sleep Apnea fs:. Hz WL:5 s Signal Time (min) J. McNames Portland State University ECE 3 Spectrograms Ver

71 Example : Arterial Blood Pressure Generate the spectrogram of an arterial blood pressure signal from a child in an intensive care unit (ICU) with sepsis. Interpret the figure. J. McNames Portland State University ECE 3 Spectrograms Ver

72 ABP (mmhg) Example : ABP Spectrogram ) Respiration Fundamental Respiration Second Harmonic Cardiac Second Harmonic ) Amplitude Cardiac Modulation Fundamental 3) Frequency Modulation Time (s) J. McNames Portland State University ECE 3 Spectrograms Ver

73 Example 7: ABP and ICP Generate the spectrogram of arterial blood pressure and intracranial pressure signals from a child with traumatic brain injury. J. McNames Portland State University ECE 3 Spectrograms Ver

74 Example 7: ABP and ICP 3 ICP (mmhg) ABP (mmhg) Time (min) J. McNames Portland State University ECE 3 Spectrograms Ver

75 Example 7: ABP and ICP ICP (mmhg) L=. s 1 ABP (mmhg) L=. s J. McNames Portland State University ECE 3 Spectrograms Ver

76 Example 7: ABP and ICP 1 1 ICP Time (min) ABP Time (min) J. McNames Portland State University ECE 3 Spectrograms Ver

77 Other Estimators Spectrograms (and their generalizations) is a topic that warrants an entire class itself There are many estimators of how power is distributed across time and frequency Includes wavelets ( scaleograms ) J. McNames Portland State University ECE 3 Spectrograms Ver

78 Summary Moving window estimates can be used to perform spectrogram analysis of locally stationary signals Window length is the most critical parameter Controls tradeoff between time and frequency resolution Duration-bandwidth product is limited by the uncertainty principle Can apply to joint estimation techniques as well Very useful for observing time-varying statistical properties J. McNames Portland State University ECE 3 Spectrograms Ver