Week3Video - tut PDF

Preview

Summary

tut...

Description

Queensland University of Technology EGB242: Signal Analysis Video Tutorial Week 3

Introduction Joseph Fourier in 1822 published his work on heat flow in Th´eorie analytique de la chaleur, and presented one of the most important contributions to mathematics: The Fourier Series. Fourier series is the decomposition of periodic functions into an infinite sum of weighted, orthogonal, and harmonically related sine and cosine basis functions. This tutorial guides you through the process of expanding periodic functions into its trigonometric Fourier Series.

Trigonometric Fourier Series The Fourier Series representation f (t) of a signal s (t) is calculated by f (t) = a0 +

∞ X

an cos (2πnf0 t) +

n=1

∞ X

bn sin (2πnf0 t) ,

(1)

n=1

where, a0 = an = bn =

1 T 2 T 2 T

Z

t0 +T

t Z 0t0 +T

t0 Z t0 +T

s (t) dt ,

(2)

s (t) cos (2πnf0 t) dt ,

(3)

s (t) sin (2πnf0 t) dt .

(4)

t0

A good way to understand the essence of Fourier series is to treat the f (t) expression as being made up of three parts, f (t) =

a0 + Average

∞ X

an cos (2πnf0 t) +

n=1

∞ X

bn sin (2πnf0 t) .

(5)

n=1 Cosine basis

Sine basis

The a0 term represents the average of s (t) over one period. Its calculation given in (2) reflects the calculation of an average. The an and bn terms give the amplitudes of the fundamental frequency and each related harmonic for the cosine (even) and sine (odd) basis functions respectively. The fundamental frequency f0 refers to the frequency at which s (t) repeats itself. The related harmonics refer to the frequencies that are integer multiples of 1

the fundamental frequecy, n × f0 . In particular, when n = 1, a1 and b1 are the coefficients for the sine and cosine basis functions at the fundamental frequency (also called the first harmonic); and when n = 2, a2 and b2 are the coefficients for the sine and cosine basis functions of the second harmonic.

Question 1 Follow the steps to compute the trigonometric Fourier series of s1 (t), given below in Fig. 1. 1

s1 (t)

0.5

0

−0.5

−1 −1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

t (seconds)

Figure 1: Waveform for s1 (t) (a) What is the period of s1 (t)? What is the fundamental frequency f0 ? (b) What is the DC component of the signal? Find also, a0 . (c) Is s1 (t) an even function? (d) Find an and bn . (e) Use the fseries.p function to plot and check your results. [f,t] = fseries(a0,an,bn,f0) The fseries function computes the expression of the Fourier series, f and its corresponding time vector over the interval [0, T ). It also generates a plot of f vs t. Function fseries accepts four inputs. a0 and f0 are to be single values, while an and bn are expected to be row vectors of equal length representing the coefficients of the cosine and sine basis functions.

2

Question 2: Follow the steps to compute the trigonometric Fourier series of s2 (t), given below in Fig. 2.

0.5 0.4 0.3 0.2

s2(t)

0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −2.5

−2

−1.5

−1

−0.5

0 0.5 t (seconds)

1

1.5

2

2.5

Figure 2: Waveform for s2 (t) (a) What is the period of s2 (t)? What is the fundamental frequency f0 ? (b) What is the DC component of the signal? Find also, a0 . (c) Is s2 (t) an odd function? (d) Find an and bn . (e) Use the fseries.p function to plot and check your results.

3

Question 3: Follow the steps to compute the trigonometric Fourier series of s3 (t), for which one period is shown below in Fig. 3. 1 0.75

s3 (t) 0.5 0.25 0 0

0.2

0.4

0.6

0.8

1

t (seconds)

Figure 3: Waveform for s3 (t) (a) What is the DC component of the signal? Find also, a0 . (b) Is s3 (t) an even or an odd function? (c) Find an and bn . (d) Use the fseries.p function to plot and check your results.

Question 4: One period of the periodic signal s4 (t), between the domain of 0 ≤ t < 1 is described by the function, s4 (t) = exp (2t) . Find its corresponding Fourier series expansion.

4

(6)...