PART 1: SYMBOLIC FOURIER TRANSFORMS (8 PTS)
1. Write MATLAB code that will find the Fourier transform 𝑭𝑭(𝝎𝝎) of each input 𝒇𝒇(𝒕𝒕). Include the final result
in your lab report using Equation Editor or MathType in Word. Remember to define your variables as
symbolic when appropriate. Simplify all MATLAB answers as much as possible.
2. Write MATLAB code that will find the inverse Fourier transform 𝒇𝒇(𝒕𝒕) of each input 𝑭𝑭(𝝎𝝎). Include the final
result in your lab report using Equation Editor or MathType in Word. Remember to define your variables as
symbolic when appropriate. Simplify all MATLAB answers as much as possible
PART 2: INVESTIGATING AUDIO RECORDINGS OF A PIANO WITH A DISCRETE FOURIER TRANSFORM (47 PTS)
In the last lab, you started to analyze an audio recording in both the time and the frequency domain. You will
continue with the analysis in this lab.
Examining a signal in the frequency domain can sometimes reveal its underlying structure more clearly than
examining it in the time domain. This is especially true when the signal contains multiple periodic components
that, when mixed together in the time domain, become difficult to separate by eye (recall your MiniLab and the
difficulty seeing 4 different frequencies in the time domain). In this question, you will use the discrete Fourier
transform to analyze audio recordings of notes played on a piano. Your analysis will be deeper and more complete
than what would be possible in the time domain alone.
1. Load the Sound Files
Load the audio recordings1
into the workspace with the following line of Matlab code:
The vectors “note1”, “note2”, and “note3” contain audio recordings of a single note being played. The vector
“chord” is a combination of three notes being played at once. The audio files are sampled at 44,100 Hz (44,100
data points per second), and the recordings contain 97,021 samples (2.2 seconds of sound). Each data point
represents the air pressure exerted on the microphone at that moment in time.
First, listen to the sounds by running the following two lines of Matlab code:
p = audioplayer(note1, fs);
To listen to the other notes and chord, replace “note1” with a different sound name.
3. Time Domain Plots (10 pts)
As a first pass to understanding the data, plot it in the time domain. Make one figure with 1×3 subplots that plot
note3 against time (with time on the x-axis), each with a different time axis: [0 2.2], [1 1.2], and [1 1.005]. (6 pts)