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Take Derivatives of a Signal

You want to differentiate a signal without increasing the noise power. MATLAB®'s function diff amplifies the noise, and the resulting inaccuracy worsens for higher derivatives. To fix this problem, use a differentiator filter instead.

Analyze the displacement of a building floor during an earthquake. Find the speed and acceleration as functions of time.

Load the file earthquake. The file contains the following variables:

  • drift: Floor displacement, measured in centimeters

  • t: Time, measured in seconds

  • Fs: Sample rate, equal to 1 kHz

load('earthquake.mat')

Use pwelch to display an estimate of the power spectrum of the signal. Note how most of the signal energy is contained in frequencies below 100 Hz.

pwelch(drift,[],[],[],Fs)

Figure contains an axes object. The axes object with title Welch Power Spectral Density Estimate contains an object of type line.

Use designfilt to design an FIR differentiator of order 50. To include most of the signal energy, specify a passband frequency of 100 Hz and a stopband frequency of 120 Hz. Inspect the filter with fvtool.

Nf = 50; 
Fpass = 100; 
Fstop = 120;

d = designfilt('differentiatorfir','FilterOrder',Nf, ...
    'PassbandFrequency',Fpass,'StopbandFrequency',Fstop, ...
    'SampleRate',Fs);

fvtool(d,'MagnitudeDisplay','zero-phase','Fs',Fs)

Figure Filter Visualization Tool - Zero-phase Response contains an axes object and other objects of type uitoolbar, uimenu. The axes object with title Zero-phase Response contains 2 objects of type line.

Differentiate the drift to find the speed. Divide the derivative by dt, the time interval between consecutive samples, to set the correct units.

dt = t(2)-t(1);

vdrift = filter(d,drift)/dt;

The filtered signal is delayed. Use grpdelay to determine that the delay is half the filter order. Compensate for it by discarding samples.

delay = mean(grpdelay(d))
delay = 25
tt = t(1:end-delay);
vd = vdrift;
vd(1:delay) = [];

The output also includes a transient whose length equals the filter order, or twice the group delay. delay samples were discarded above. Discard delay more to eliminate the transient.

tt(1:delay) = [];
vd(1:delay) = [];

Plot the drift and the drift speed. Use findpeaks to verify that the maxima and minima of the drift correspond to the zero crossings of its derivative.

[pkp,lcp] = findpeaks(drift);
zcp = zeros(size(lcp));

[pkm,lcm] = findpeaks(-drift);
zcm = zeros(size(lcm));

subplot(2,1,1)
plot(t,drift,t([lcp lcm]),[pkp -pkm],'or')
xlabel('Time (s)')
ylabel('Displacement (cm)')
grid

subplot(2,1,2)
plot(tt,vd,t([lcp lcm]),[zcp zcm],'or')
xlabel('Time (s)')
ylabel('Speed (cm/s)')
grid

Figure contains 2 axes objects. Axes object 1 contains 3 objects of type line. Axes object 2 contains 3 objects of type line.

Differentiate the drift speed to find the acceleration. The lag is twice as long. Discard twice as many samples to compensate for the delay, and the same number to eliminate the transient. Plot the speed and acceleration.

adrift = filter(d,vdrift)/dt;

at = t(1:end-2*delay);
ad = adrift;
ad(1:2*delay) = [];

at(1:2*delay) = [];
ad(1:2*delay) = [];

subplot(2,1,1)
plot(tt,vd)
xlabel('Time (s)')
ylabel('Speed (cm/s)')
grid

subplot(2,1,2)
plot(at,ad)
ax = gca;
ax.YLim = 2000*[-1 1];
xlabel('Time (s)')
ylabel('Acceleration (cm/s^2)')
grid

Figure contains 2 axes objects. Axes object 1 contains an object of type line. Axes object 2 contains an object of type line.

Compute the acceleration using diff. Add zeros to compensate for the change in array size. Compare the result to that obtained with the filter. Notice the amount of high-frequency noise.

vdiff = diff([drift;0])/dt;
adiff = diff([vdiff;0])/dt;

subplot(2,1,1)
plot(at,ad)
ax = gca;
ax.YLim = 2000*[-1 1];
xlabel('Time (s)')
ylabel('Acceleration (cm/s^2)')
grid
legend('Filter')
title('Acceleration with Differentiation Filter')

subplot(2,1,2)
plot(t,adiff)
ax = gca;
ax.YLim = 2000*[-1 1];
xlabel('Time (s)')
ylabel('Acceleration (cm/s^2)')
grid
legend('diff')

Figure contains 2 axes objects. Axes object 1 with title Acceleration with Differentiation Filter contains an object of type line. This object represents Filter. Axes object 2 contains an object of type line. This object represents diff.

See Also

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