Digital Signal Processing using the Fast Fourier Transform (FFT)

Department of Aeronautical and Automotive Engineering and
Transport Studies
MSc in Automotive Systems Engineering
Module 1: Engineering Framework
Course :
Signal Analysis

 Coursework: Digital Signal Processing
using the Fast Fourier Transform

by: Albert Kaiser
date: June 1997 (update 2005)

Summary

Conventionally a signal is a physical variable that changes with time and contains information. The signal may be represented in analogue (continuos) or discrete (digital) form. The majority of the physical variables of interest for the engineer are of analogue form. However digital data acquisition equipment favour a digital representation of the analogue signal.

The digital representation of a analogue signal will effect the characteristic of the signal. Thus an understanding of the underlying principles involved in signal processing is essential in order to retain the basic information of the original signal.

The primary goal to use the Discrete Fourier Transform (DFT) is to approximate the Fourier Transform of a continuous time signal. The DFT is discrete in time and frequency domain and has two important properties:

• the DFT is periodic with the sampling frequency
• the DFT is symmetric about the Nyquist frequency

Due to the limitations of the DFT there are three possible phenomena that could result in errors between computed and desired transform.

• Aliasing
• Picket Fence Effect
• Leakage

The DFT of a signal uses only a finite record length of the signal. Thus the input signal for the DFT can be considered as the result of multiplying the signal with a window function. Multiplication in the time domain results in convolution in the frequency domain, which will influence the spectral characteristic of the sampled signal. In the table below rectangular and Hanning window are compared:

[...] Table

The Fast Fourier Transform (FFT) is a computationally efficient algorithm for evaluating the DFT of a signal. It is imported to appreciate the properties of the FFT if it is to be used effectively for the analysis of signals. In order to avoid aliasing and resulting misinterpretation of measurement data the following steps should be followed:

[...]

Table of Contents

1 OBJECTIVE ... 5

2 APPROACH  ... 5

3 INTRODUCTION ... 5

4 ANALYTICAL  ... 6
4.1 INTRODUCTION ... 6
4.1.1 Classification of Signals  ... 6
4.1.2 Periodic Signals  ... 6
4.2 FOURIER SERIES ... 6
4.2.1 Fourier Series of a square wave ... 7
4.3 FOURIER TRANSFORM ... 8
4.3.1 Definition of the Fourier Transform ... 8
4.3.2 Fourier Transform of a sin wave ... 9
4.3.3 Fourier Transform of a square wave  ... 9
4.3.4 Fourier Transform of a rectangular pulse  ... 10
4.3.5 Window Functions  ... 10
4.3.5.1 Convolution in Frequency Domain ... 10
4.3.5.2 Convolution in Time Domain ... 11
4.3.5.3 Fourier Transform of a rectangular window ... 12
4.3.5.4 Fourier Transform of a Hanning Window ... 12
4.3.5.5 Periodic Function through Rectangular Window ... 13
4.3.5.6 Periodic Function through Hanning Window ... 14
4.4 DISCRETE FOURIER TRANSFORM ... 16
4.4.1 Definition of the Discrete Fourier Transform  ... 16
4.4.2 The DFT spectrum is periodic in Frequency  ... 16
4.4.3 The DFT spectrum is symmetric about the in Nyquist frequency  ... 17
4.4.4 Practical Considerations  ... 17
4.4.4.1 Aliasing ... 17
4.4.4.2 Picket Fence Effect  ... 18
4.4.4.3 Leakage ... 18
4.5 FAST FOURIER TRANSFORMATION  ... 18

5 APPLICATION OF THE FFT ... 19
5.1 SIN-WAVE ... 19
5.2 TWO SUPERIMPOSED SIN-WAVES ... 20
5.3 TWO SUPERIMPOSED SIN-WAVES WITH ADDED RANDOM NOISE ... 21
5.4 SQUARE WAVE  ... 22
5.5 NARROW PULSE ... 23
5.6 BROAD PULSE ... 24
5.7 RECTANGULAR WINDOW ... 25
5.8 HANNING WINDOW  ... 26
5.9 SINE WAVE TROUGH RECTANGULAR AND HANNING WINDOW  ... 27
5.10 LEAKY SINE WAVE TROUGH RECTANGULAR AND HANNING WINDOW ... 28
5.11 SINE WAVE TROUGH NARROW RECTANGULAR AND HANNING WINDOW  ... 29

6 CONCLUSION ... 30

7 REFERENCE ... 31

8 APPENDIX  ... 31

[...]