**I’m attempting to thoroughly re-write the data assessment** portions of my QRSS VD software, and rather than rushing to code it (like I did last time) I’m working hard on every step trying to optimize the code. I came across some notes I made about Fast Fourier Transformations from the first time I coded the software, and though I’d post some code I found helpful. Of particular satisfaction is an email I received from Alberto, I2PHD, the creator of Argo (the “gold standard” QRSS spectrograph software for Windows). In it he notes:

I think that [it is a mistake to] throw away the imaginary part of the FFT. What I do in Argo, in Spectran, in Winrad, in SDRadio and in all of my other programs is compute the magnitude of the [FFT] signal, then compute the logarithm of it, and only then I do a mapping of the colors on the screen with the result of this last computation.

**These concepts are simple** to visualize when graphed. Here I’ve written a short Python script to listen to the microphone (which is being fed a 2kHz sine wave), perform the FFT, and graph the real FFT component, imaginary FFT component, and their sum. The output is:

**Of particular interest** to me is the beautiful complementarity of the two curves. It makes me wonder what types of data can be extracted by the individual curves (or perhaps their difference?) down the road. I wonder if phase measurements would be useful in extracting weak carries from beneath the noise floor?

**Here’s the code I used to generate the image above.** Note that my microphone device was set to listen to my stereo output, and I generated a 2kHz sine wave using the command `speaker-test -t sine -f 2000`

on a PC running Linux. I hope you find it useful!

import numpy import pyaudio import pylab import numpy ### RECORD AUDIO FROM MICROPHONE ### rate=44100 soundcard=1 #CUSTOMIZE THIS!!! p=pyaudio.PyAudio() strm=p.open(format=pyaudio.paInt16,channels=1,rate=rate, input_device_index=soundcard,input=True) strm.read(1024) #prime the sound card this way pcm=numpy.fromstring(strm.read(1024), dtype=numpy.int16) ### DO THE FFT ANALYSIS ### fft=numpy.fft.fft(pcm) fftr=10*numpy.log10(abs(fft.real))[:len(pcm)/2] ffti=10*numpy.log10(abs(fft.imag))[:len(pcm)/2] fftb=10*numpy.log10(numpy.sqrt(fft.imag**2+fft.real**2))[:len(pcm)/2] freq=numpy.fft.fftfreq(numpy.arange(len(pcm)).shape[-1])[:len(pcm)/2] freq=freq*rate/1000 #make the frequency scale ### GRAPH THIS STUFF ### pylab.subplot(411) pylab.title("Original Data") pylab.grid() pylab.plot(numpy.arange(len(pcm))/float(rate)*1000,pcm,'r-',alpha=1) pylab.xlabel("Time (milliseconds)") pylab.ylabel("Amplitude") pylab.subplot(412) pylab.title("Real FFT") pylab.xlabel("Frequency (kHz)") pylab.ylabel("Power") pylab.grid() pylab.plot(freq,fftr,'b-',alpha=1) pylab.subplot(413) pylab.title("Imaginary FFT") pylab.xlabel("Frequency (kHz)") pylab.ylabel("Power") pylab.grid() pylab.plot(freq,ffti,'g-',alpha=1) pylab.subplot(414) pylab.title("Real+Imaginary FFT") pylab.xlabel("Frequency (kHz)") pylab.ylabel("Power") pylab.grid() pylab.plot(freq,fftb,'k-',alpha=1) pylab.show()

**After fighting for a while long with** a “shifty baseline” of the FFT, I came to another understanding. Let me first address the problem. Taking the FFT of different regions of the 2kHz wave I got traces with the peak in the identical location, but the “baselines” completely different.

**Like many things, I re-invented the wheel.** Since I knew the PCM values weren’t changing, the only variable was the starting/stopping point of the linear sample. “Hard edges”, I imagined, must be the problem. I then wrote the following function to shape the PCM audio like a triangle, silencing the edges and sweeping the volume up toward the middle of the sample:

def shapeTriangle(data): triangle=numpy.array(range(len(data)/2)+range(len(data)/2)[::-1])+1 return data*triangle

**After shaping the data BEFORE I applied the FFT,** I made the subsequent traces MUCH more acceptable. Observe:

**Now that I’ve done all this experimentation/thinking,** I remembered that this is nothing new! Everyone talks about shaping the wave to minimize hard edges before taking the FFT. BAH! Another case of me re-inventing the wheel because I’m too lazy to read others’ work. However, in my defense, I learned a lot by trying all this stuff — far more than I would have learned simply by copying someone else’s code into my script. Experimentation is the key to discovery!