MATLAB: Sound sample too short to see higher frequencies

fft

High Frequency FFT
Hi,
I have a Wav file with a length of approximately 292kB at 88bt/sec, for an old lister engine. I want to change the variable FreqMax to look at frequencies up to the rated RPM (around 1640), but am not able to do this because I don’t have enough data points from this file. (I understand that for 88kbps, my nyquist frequency will not be high enough to get too information, but I want to anyway.) I have three ideas about how to accomplish this but I am unsure how to implement them:
1.  Reduce the frequency resolution
2.  Drop the lower frequencies, to shift the range toward the higher frequencies
3.  Zero-pad the original sample so that I have more data points.
Are there better solutions?
I am curious how (1) or (2) can be accomplished, I am unsure how to do the scaling and cannot find good resources on this, and also if there is some better approach to looking at frequencies with this signal.
Thanks, Chris
%lister engine sound source
%http://www.oldengineshed.com/diesel.html
%3 3/16" Bore X 3 1/2" Stroke - Rated H.P. at 1650 R.P.M.
%FULL SOUND FILE
[x1 fs1 nbits1 opts1]=wavread('listersl.wav');
t1=(0:length(x1)-1)/fs1;
subplot(5,1,1);
plot(t1,x1)
legend('Full Dataset Waveform');
xlabel('t (s)');
ylabel('Amplitude');
%ANALYSIS OF SAMPLE WINDOW
%file bit rate
bitrate=88000/8; %bit/s (8bit/byte)
SampleStart=10; %what second in the file to start reading
SampleEnd=18;
WavSampleLength=[SampleStart*bitrate+1 SampleEnd*bitrate];
[x fs]=wavread('listersl.wav', WavSampleLength);
t=(SampleStart:1/fs:SampleEnd);
dif=length(t)-length(x);
t=(SampleStart:1/fs:SampleEnd-1/fs*dif);
subplot(5,1,2);
plot(t,x)
legend('Sample Waveform');
xlabel('t (s)');
ylabel('Amplitude');
xwin=x.*hamming(length(x));
subplot(5,1,3)
plot(t,xwin)
legend('Window');
xlabel('t (s)');
ylabel('Amplitude');
Y=fft(xwin);
FreqMax=10000;
hz=FreqMax*length(Y)/fs;
f=(0:hz)*fs/length(Y);
subplot(5,1,4);
plot(f,20*log10(abs(Y(1:length(f)))+eps));
legend('Spectrum');
xlabel('Freq (Hz)')
ylabel('Magnitude (dB)')
C=fft(log(abs(Y)+eps));
f1=fs/1000; %max for plot, fs/1000~1 milliseconds
f2=fs/1; %min for plot, fs/50~20 milliseconds
q=(f1:f2)/fs;
subplot(5,1,5);
plot(q,abs(C(f1:f2)));
legend('Cepstrum')
xlabel('Quefrequency (s)')
ylabel('Amplitude')

Best Answer

The Nyquist frequency is, by definition, the highest frequency you can resolve before you start aliasing the signal. This means the amplitude you derive for any frequency beyond the Nyquist will likely have little meaning. You find the Nyquist by:
sample_rate = 20; % units = samples per second;
nyquist_freq = sample_rate / 2; % = 10 Hz
Notice this equation has nothing to do with how long the timeseries is. The length of timeseries along with the sample rate controls the frequency increment (the smallest resolvable frequency) like this:
sample_rate = 20;
num_samples = 1024;
time_increment = 1 / sample_rate;
freq_increment = sample_rate / num_samples;
The fact is, there is no way to recover or resolve higher frequencies unless you have a higher sample rate. Period. If you can't increase the sample rate of your measuring device and you want bogus results, then you can interpolate between the points in your time series to increase the sample rate. You need to interpolate to a sample rate that pushes the Nyquist frequency beyond the highest frequency you will be resolving bogus-ly.
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