Using the Fourier transform (or Fourier series), signals can be decomposed into weighted sums of sines and cosines. If we examine the defining equations for Fourier analysis they almost resemble a convolution or correlation. In this laboratory exercise we will explore this similarity.

1. A synthetic audio signal s has been generated and is displayed below.¹ You can listen to it by pressing the s control (). This might involve the adjustment of the volume of your device.

In the top panel we show the sinusoidal matched-filter h and a synthetic audio signal s. The matched filter h has a frequency (ω = 2πf) and a finite duration (width) as shown below. The finite duration can be thought of as a window on an infinite-duration sinusoidal signal. The topic of “windows” will be dealt with extensively in Chapter 12

In the bottom panel we show the noise-contaminated synthetic signal r=s+N where N is additive, independent, Gaussian noise. The output of the matched filter is φ_rh.

There are controls for choosing the window width, the SNR, and the frequency f. By changing the frequency f, you can see in the bottom panel which sections of the audio signal r=s+N have a strong “match”—a strong correlation—with the sinusoidal matched filter.

A formal Fourier analysis involves both an even function, the cosine, and an odd function, the sine. Further, a Fourier spectrum requires a summation (or integration) interval from –∞ to +∞. Nevertheless, we can use the sine function to determine a “short-time spectrum”.

2. Does the use of a sine function represent a serious constraint? Do we miss the spectral components that might be found with a cosine function?


3. How does the result of the matched filtering change as we increase the frequency f.


4. By listening to the synthetic audio signal and looking at the output of the matched filter, estimate the frequency components of the audio signal.


5. How do the results change as you increase the number of samples in the matched filter window?


6. How does this short-time Fourier analysis change as the SNR is decreased, that is, as the noise level is increased? At what SNR is the spectrum of the tones no longer recognizable? At this SNR are the tones still audible to you?


7. Explain the hexagonal envelope of the matched filter output φ_rh when the filter window is slightly less than a tone duration and the frequency f is close to the tone frequency.

¹Each of the signals and correlation functions contains far more samples than can be displayed across the screen of your device. The data are therefore “binned” to fit on the screen. Slight distortions—artefacts of the binning process—are therefore possible. ↵