# Insights Into FFTs, Imaginary Numbers, and Accurate Spectrographs

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**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.

Alberto, I2PHD (the creator of Argo)

UPDATE IN SEPTEMBER, 2020 (10 years later):I now understand that`magnitude = sqrt(real^2 + imag^2)`

and this post is a bit embarrassing to read! Check out my .NET FFT library FftSharp for a more advanced discussion on this topic.

**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 complementary 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. They call it *windowing.* 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!