Introduction

The objective of this lab is to use computers to analyze the spectra of real-world signals. The Discrete Fourier Transform (DFT), which is the discrete-time version of the Fourier series, is the main tool we will be using. Once a signal is sampled, the spectrum can be analyzed by applying the DFT to the samples. The DFT can be rapidly computed using an algorithm known as the Fast Fourier Transform (FFT). Mat lab has a built-in function, FFT, that implements this algorithm. The FFT algorithm was discovered by Gauss and later rediscovered and popularized by Cooley and Tukey in the late 1960's. The ability to perform spectrum analysis on a computer revolutionized science and engineering, and the FFT is one of the most important tools in signal processing and computational science. In fact, you have all encountered the FFT before; every time you listen to an MP3 file or view a JPEG image you are using an FFT-based algorithm.

What is the frequency of the sinusoid that was sampled? Was it adequately sampled, at or above the Nyquist rate?

How many samples were acquired per cycle of the sinusoid?

Lab Exercise

Mathematically explain why the spectrum of the 150Hz sinusoid in the warm-up did not consist of two perfect peaks. This effect is often called "spectral leakage".