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Time Frequency Analysis Overview Introduction and Motivation Introduction and motivation r x ( , n ) = E [ x ( n ) x ( n ) ] for m m Windowed estimates as filter banks R x (e j , n ) = r x ( , n )

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SLIDE 1

Practical Application

  • In practice, time-frequency analysis is most frequently performed

by estimating the statistical properties from sliding windows

  • Resulting estimates, e.g., R(ejω, n), are a function of two

independent variables

  • 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 538/638 Time Frequency Analysis

  • Ver. 1.01

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Time Frequency Analysis Overview

  • Introduction and motivation
  • Windowed estimates as filter banks
  • Uncertainty principle
  • Examples
  • Other estimates
  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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Example 1: Chirp Spectrogram Apply the modified periodogram on a chirp signal with a linear frequency sweep from 1–10 Hz over a period of 20 s.

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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Introduction and Motivation rx(ℓ, n) = E [x(n)x(n − ℓ)∗] for m ≤ ℓ ≤ m Rx(ejω, n) =

∞

  • ℓ=−∞

rx(ℓ, n) ≈

m

  • ℓ=−m

rx(ℓ, n)

  • In practical applications, many signals are nonstationary
  • In order to apply the techniques, we usually assume the signal

statistics change “slowly”

  • “Locally” stationary
  • Rate of change depends on the process
  • Time frequency analysis tries to estimate how these properties

change with time

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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SLIDE 2

Example 1: Chirp Spectrogram

Chirp Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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Example 1: Chirp Spectrogram

Chirp Signal Tw= 0.2 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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Example 1: Chirp Spectrogram

Chirp Signal Tw=15.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

8

Example 1: Chirp Spectrogram

Chirp Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

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SLIDE 3

Why the Windowed Periodogram?

  • Why not a smoother technique with less variance?

– The eye can smooth the plots visually – Techniques to decrease variance decrease the resolution too much

  • 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

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Example 1: Relevant MATLAB Code

fs = 20; % Sample rate (Hz) T = 20; % Signal duration (s) f = [1 10]; % Chirp frequencies [start stop] Tw = [0.2 0.5 2 5 15]; % Window lengths k = 0:T*fs; % Discrete time index t = (k-0.5)/fs; % Sample times (s) x = chirp(t,f(1),T,f(2)); for c1=1:length(Tw), tw = Tw(c1); [S,t,f] = NonparametricSpectrogram(x,fs,tw); figure; I = abs(S).^2; cr = prctile(reshape(I,prod(size(I)),1),[0.5 99.5]); ha = axes(’Position’,[0.14 0.20 0.80 0.73]); h = imagesc(t,f,abs(S).^2,cr); hold on; h = plot(tw/2*[1 1],[0 fs/2],’k’,tw/2*[1 1],[0 fs/2],’w’); set(h(1),’LineWidth’,2); set(h(2),’LineWidth’,1); h = plot((T-tw/2)*[1 1],[0 fs/2],’k’,(T-tw/2)*[1 1],[0 fs/2],’w’); set(h(1),’LineWidth’,2); set(h(2),’LineWidth’,1); hold off;

  • J. McNames

Portland State University ECE 538/638 Time Frequency Analysis

  • Ver. 1.01

9

Filter Bank Interpretation X(ejω, n) =

∞

  • ℓ=−∞

x(n + ℓ)w(ℓ)e−jωℓ =

∞

  • m=−∞

x(m)w(m − n)e−jω(m−n) =

∞

  • m=−∞

x(m)w(−(n − m))ejω(n−m) = x(n) ∗ w(−n)ejωn ˆ R(MP)(ejω, n) = 1 L|X(ejω, n)|2

  • The spectrogram can be thought of as the squared magnitude of

the output of an LTI system with impulse response hω(n) = w(−n)ejωn

  • The magnitude response evaluated at ωo is then W
  • ej(ωo−ω)
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Example 1: Relevant MATLAB Code Continued

set(gca,’YDir’,’Normal’); set(gca,’XAxisLocation’,’Top’); set(gca,’YAxisLocation’,’Right’); xlim([0 T]); ylim([0 fs/2]); ht = title(sprintf(’Chirp Signal T_w=%4.1f s’,tw)); ha = axes(’Position’,[0.14 0.10 0.80 0.09]); h = plot((k-0.5)/fs,x,’b’); set(h,’LineWidth’,0.5); xlim([0 T]); set(ha,’YTick’,[]) xlabel(’Time (s)’); ha = axes(’Position’,[0.07 0.20 0.065 0.73]); Rx = mean(I,2); h = plot(Rx,f,’r’); set(h,’LineWidth’,0.5); ylim([0 fs/2]); set(ha,’XTick’,[]) ylabel(’Frequency (Hz)’); set(ht,’VerticalAlignment’,’Top’) end;

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SLIDE 4

Filter Bank Frequency Responses with Padding

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 Frequency (radians/sample) |W(ej ω)| FFT Filters Blackman Window L=17 P=32

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Filter Bank Frequency Responses without Padding

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 Frequency (radians/sample) |W(ej ω)| FFT Filters Blackman Window L=17 P=17

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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, L
  • 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

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Effect of Zero Padding

20 40 60 80 100 120 −1 −0.5 0.5 Time Domain 16 points 0.1 0.2 0.3 0.4 1 2 3 Frequency Domain 20 40 60 80 100 120 −1 −0.5 0.5 128 points (112 zeros) 0.1 0.2 0.3 0.4 1 2 3 Frequency Domain

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SLIDE 5

Uncertainty Principle It can be shown that σt,x × σω,x ≥ 1 2

  • 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

  • Not aware of an equivalent uncertainty principle for DT signals
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Energy Density Functions

  • Note that the windowed signal segments have finite energy
  • Analogous ideas apply in continuous time
  • Consider, for a moment, a continuous time windowed segment

x(t)

  • Defined the normalized temporal energy density function and

spectral energy density function as ex(t) = |x(t)|2 ∞

−∞ |x(t)|2 dt

Ex(ω) = |X(jω)|2 ∞

−∞ |X(jω)|2 dω

  • Like pdfs, both of these functions are nonnegative and have unit

area

  • Can be thought of as energy density functions (edfs) in time and

frequency

  • Like pdfs, we can define energy moments
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Time Bandwidth Product

−100 −50 50 100 −1 1 Windowed Sinusoid ω0=0.628 rad/sample L=33 1 2 3 5 10 Frequency Domain X(ejω −100 −50 50 100 −1 1 L=65 1 2 3 10 20 X(ejω −100 −50 50 100 −1 1 Sample Index (n) L=129 1 2 3 50 Frequency (rad/sec) X(ejω

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Bandwidth µt,x = ∞

−∞

t ex(t) dt µω,x = ∞

−∞

ω Ex(ω) dω σ2

t,x =

∞

−∞

|t − µt,x|2ex(t) dt σ2

ω,x =

∞

−∞

|ω − µω,x|2Ex(ω) dω

  • There are many definitions of duration and bandwidth of a signal
  • One pair of possible definitions is based on the second-order

moments of the energy density functions

  • Let us define the signal

– Duration as the standard deviation of the temporal edf, σt,x – Bandwidth as the standard deviation of the spectral edf, σω,x

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SLIDE 6

MATLAB Code Continued

subplot(nL,2,c1*2-1); h = plot(n,x); xlim([-max(L) max(L)]); ylim([ymn-0.05*yrg ymx+0.05*yrg]); AxisLines; box off; ylabel(sprintf(’L=%d’,l)); if c1==1, title(sprintf(’Windowed Sinusoid \\omega_0=%5.3f rad/sample’,w0)); elseif c1==nL, xlabel(’Sample Index (n)’); end; subplot(nL,2,c1*2); h = plot(w(kw),X(kw),’r’); xlim([0 pi]); ylim([0 1.05*max(X)]); AxisLines; ylabel(’X(e^{j\omega}’); box off; if c1==1, title(’Frequency Domain’); elseif c1==nL, xlabel(’Frequency (rad/sec)’); end; end;

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MATLAB Code

NZ = 2^10; w0 = pi/5; L = [32 64 128]+1; nL = length(L); k = (0:NZ-1)’; w = k*2*pi/NZ; kw = 1:NZ/2; l = 1001; wn = blackman(l); wn = wn*sqrt(l/sum(wn.^2)); n = (-(l-1)/2:(l-1)/2)’; x = cos(w0*n).*wn; ymx = max(x); ymn = min(x); yrg = ymx-ymn;

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Attaining the Uncertainty Principle The Gaussian function attains the uncertainty principle with equality x(t) =

1 √ 2πσ2 e− t2

2σ2

X(jω) = e− ω2σ2

2

ex(t) = 1 √πσ e− t2

σ2

Ex(ω) = 1 √ πσ−1 e−ω2σ2 σt,x = σ √ 2 σω,x = 1 √ 2σ σt,xσω,x = 1 2

  • If this has the best duration-bandwidth product (i.e., resolution),

why don’t we use this as a window?

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MATLAB Code Continued

for c1=1:nL, l = L(c1); n = (-(l-1)/2:(l-1)/2)’; wn = blackman(l+2); wn = wn(2:end-1); % Trim off the zeros wn = wn*sqrt(l/sum(wn.^2)); % Scale appropriately x = cos(w0*n).*wn; % Windowed signal segment X = exp(j*(l-1)/2*w).*fft(x,NZ); % Calculate frequency response %disp(max(imag(X))); X = real(X); % Eliminate imaginary part due to finite resolution

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SLIDE 7

Example 2: Cosine Spectrogram

Cosine Signal Tw= 0.2 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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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.)

  • Often easier to visualize the square root of the PSD, rather than

the PSD

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Example 2: Cosine Spectrogram

Cosine Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Cosine, Impulse, and Step Spectrograms Repeat the previous example for a 5 Hz sinusoidal signal, the unit impulse at t = 10 s, and the unit step starting at t = 10 s.

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SLIDE 8

Example 2: Impulse Spectrogram

Impulse Signal Tw= 0.2 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Cosine Spectrogram

Cosine Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Impulse Spectrogram

Impulse Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Cosine Spectrogram

Cosine Signal Tw=10.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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SLIDE 9

Example 2: Unit Step Spectrogram

UnitStep Signal Tw= 0.2 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Impulse Spectrogram

Impulse Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Unit Step Spectrogram

UnitStep Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 2: Impulse Spectrogram

Impulse Signal Tw=10.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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SLIDE 10

Periodic Signals

  • Periodic signals consist of frequency components only at multiples
  • f 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

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Example 2: Unit Step Spectrogram

UnitStep Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 3: Impulse Train Spectrograms Repeat the previous example for a 1 Hz impulse train.

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Example 2: Unit Step Spectrogram

UnitStep Signal Tw=10.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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SLIDE 11

Example 3: Impulse Train Spectrogram

ImpulseTrain Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 3: Impulse Train Spectrogram

ImpulseTrain Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 3: Impulse Train Spectrogram

ImpulseTrain Signal Tw= 5.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 3: Impulse Train Spectrogram

ImpulseTrain Signal Tw= 1.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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SLIDE 12

Example 4: White Noise Spectrograms Repeat the previous example for a white noise signal.

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Example 3: Impulse Train Spectrogram

ImpulseTrain Signal Tw=10.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 4: White Noise Spectrogram

WhiteNoise Signal Tw= 0.5 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Random Signals

  • It’s important to be able to distinguish significant features from

random fluctuations due to estimation error

  • Recall that the periodogram had excessive variance
  • Helpful to examine the spectrograms of white noise
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SLIDE 13

Example 4: White Noise Spectrogram

WhiteNoise Signal Tw= 5.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 4: White Noise Spectrogram

WhiteNoise Signal Tw= 1.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 4: White Noise Spectrogram

WhiteNoise Signal Tw=10.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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Example 4: White Noise Spectrogram

WhiteNoise Signal Tw= 2.0 s 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 2 4 6 8 10 12 14 16 18 20 Time (s) 2 4 6 8 10 Frequency (Hz)

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SLIDE 14

Example 5: Apnea Detection

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Frequency (Hz) 50 100 150 200 250 0.6 0.8 1 1.2 1.4 1.6 Time (min) Signal Sleep Apnea fs: 0.8 Hz WL:256 s 50 100 150 200 250 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

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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 2000
  • Data set

– 25 recordings labelled by expert – 10 unlabelled recording – Approximately 8 hours each

  • Objective: detect OSA for every minute of the 10 recordings
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Example 6: 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.

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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!

– 92.6% of minutes classified “correctly” – Third attempt

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SLIDE 15

Example 7: ABP and ICP

305 310 315 320 325 330 335 340 345 350 355 360 15 20 25 30 ICP (mmHg) 305 310 315 320 325 330 335 340 345 350 355 360 60 80 100 120 Time (min) ABP (mmHg)

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Example 6: ABP Spectrogram

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency (Hz) 1) Respiration Fundamental Respiration Second Harmonic Cardiac Fundamental Cardiac Second Harmonic 2) Amplitude Modulation 3) Frequency Modulation 50 100 150 200 250 1 2 3 4 5 50 100 150 200 250 22.0 36.0 50.0 Time (s) ABP (mmHg)

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Example 7: ABP and ICP

0.5 1 1.5 2 2.5 200 400 600 800 ICP (mmHg) L=60.000 s 0.5 1 1.5 2 2.5 10 Frequency (Hz) 0.5 1 1.5 2 2.5 500 1000 ABP (mmHg) L=60.000 s 0.5 1 1.5 2 2.5 10 Frequency (Hz)

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Example 7: ABP and ICP Generate the cohereogram of arterial blood pressure and intracranial pressure signals from a child with traumatic brain injury.

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SLIDE 16

Other Estimators

  • Time-frequency analysis 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”)
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Example 7: ABP and ICP

Time (min) Frequency (Hz) ICP 305 310 315 320 325 330 335 340 345 350 355 360 1 2 Time (min) Frequency (Hz) ABP 305 310 315 320 325 330 335 340 345 350 355 360 1 2

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Summary

  • Moving window estimates can be used to perform time-frequency

analysis of locally stationary signals

  • Conceptually, is the output of a bank of overlapping bandpass

filters

  • 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
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Example 7: ABP and ICP

0.2 0.4 0.6 0.8 1 00.51 0.5 1 1.5 2 2.5 Frequency (Hz) 15 20 25 30 x1 10 20 30 40 50 50 100 150 x2 Time (min) Cohereogram 10 20 30 40 50

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