The purpose of this assignment is to learn about how the duration of a signal affects the properties of its Fourier Transform.

Part 1: In Matlab, create the following signal x(t) with a duration of 50 seconds.

x(t) = cos(2π · 1 · t) + cos(2π · 2 · t)

Use myFFT to generate the Fourier Transform of x(t). Plot the absolute value of the Fourier Transform and zoom in to the range 0 to 10Hz. Can you see the two peaks corresponding to the cosines at frequencies f = 1Hz and f = 2Hz?

Now repeat the above steps but reduce the signal duration first to 5 seconds, then to 1 second. How does the Fourier Transform change as a result of shortening the signal duration? Can you think of why this might be happening (you might want to look back at Computer Assignment 1 for some insight)? If you are supposed to analyze the frequency content of a signal, would you prefer that the signal be long or short? Why?

Part 2: For signals where the frequencies vary with time, we typically use a tool called a spectrogram to analyze the signal. The spectrogram basically takes the signal, splits it down into windows, and takes the Fourier Transform of the signal in each window. In this part, you will be taking the spectrogram of a signal and evaluating how the window duration affects the frequency analysis.

Use your results from Part 1 to inform your observations.

From the Blackboard page, download mySpectrogram.m as well as the audio files piano.wav and sax.wav. Use the wavread command to load the piano signal into Matlab. Recall that you can use the soundsc function to listen to the imported sound. Use mySpectrogram to look at the frequency content. That command can be called by entering mySpectrogram(s,win,fs) where s is the imported signal, win is the window length (in seconds) and fs is the sampling rate of the signal. Create the spectrogram using window durations of 0.01, 0.1, and 0.25 seconds. How do the spectrograms differ? Why? What are the tradeoffs that an engineer should consider when selecting a spectrogram window duration? Although the piano sound is a bit easier to interpret, you might also consider loading and analyzing the sax sound since it has lots of short notes instead of a handful of long ones.

One more helpful hint: you can zoom in on your spectrogram plots by either using the magnifying glass icon in the plot window, or by using the xlim and ylim commands