MEMS 431 (FL11) Lab 3

Periodic Signals and Fourier Analysis

Objective

To understand the frequency content of periodic signals, and how to estimate it from discrete (digital) data.

Background

Any periodic function can be expressed as a Fourier series: a sum of sine and cosine waves at multiples of the fundamental frequency. The coefficients in the Fourier series completely describe the function; this set of Fourier coefficients is the “frequency-domain” representation of the function. They can be computed from well-known integral formulas. Non-periodic functions can be transformed into the frequency domain by Fourier transform.

Experimental data is often obtained by digitally sampling a continuous signal. The Fast Fourier Transform (FFT) is a discrete approximation to the Fourier transform. FFTs are used to compute a power spectral density (PSD) function. The PSD measures how much of the power in a signal is associated with a given frequency. It is important to sample a continuous signal at a high enough rate to capture fast variations (high frequency components). Sampling rate determines the bandwidth in the frequency domain. It is also important to sample the continuous signal for long enough to capture slow changes (low frequency components).

Reading

Review textbook sections on Fourier series and Fourier transforms.

Procedure

Step 1: Sine wave

Set the function generator to sine wave. Connect the output of the function generator to the input channel of the SigLab signal analyzer. Start the SigLab Spectrum Analysis software (SigLab/VSA/ME417lab3.vsa) for Lab 3. Note the bandwidth (should be 50Hz) and record length (should be 512 samples) of signal acquisition. Make sure you have selected "AA filters off".

a. Acquire a 10 Hz sine wave with the SigLab. Make sure you record multiple cycles of oscillation. Record the amplitude and frequency of the major peak in the PSD. (Click on the "^" symbol to find peaks.)

b. Increase the frequency of the sine wave to 20 Hz. Acquire and plot the spectrum. Record amplitude and frequency of the peak. Note any features of the wave and PSD that change as frequency is increased. Repeat at sine wave frequencies of 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 Hz. What does the signal look like? What happens to the peak in the PSD?

c. Increase the acquisition bandwidth to 200 Hz and reacquire the 120 Hz sine wave. What do the signal (time domain) and PSD (frequency domain) look like now?

Step 2: Square wave

Switch to a square wave. Decrease the square wave frequency (cycles/sec, or Hz) to 10 Hz. Keep the acquisition bandwidth at 200 Hz.

a. Acquire a square wave with the SigLab Spectrum Analyzer. Make sure you acquire multiple cycles of the wave. Note the features of the time domain signal and the PSD.

b. Record the amplitudes and frequencies of peaks in the PSD. (Click on the "^" symbol to find peaks.)

Step 3: Triangular wave

Switch to a triangular wave (10 Hz).

a. Acquire a triangular wave with the SigLab spectrum analyzer. Make sure you acquire multiple cycles of the wave. Note the features of the time domain signal and the PSD.

b. Record the amplitudes and frequencies of peaks in the PSD. (Click on the "^" symbol to find peaks.)

Report

There is no specific format required for the report, but the following topics should be addressed:

a. THEORY Compute the Fourier series representations of the square wave, the triangular wave, and the sine wave.

b. EXPERIMENT (1) Describe what happens to (a) the time domain signal and (b) the frequency domain spectrum when the frequency of a signal exceeds the specified acquisition bandwidth. This is called aliasing. (2) Tabulate the amplitudes and frequencies of peaks in the PSD for each experiment above (sine wave, square wave, triangular wave). Compare them with the theoretical values of the Fourier coefficients. What is the relationship?