Chapter 4. Fourier Analysis for Con5nuous-Time Signals and Systems Chapter Objec5ves

Transcription

1 Chapter 4. Fourier Analysis for Con5nuous-Time Signals and Systems Chapter Objec5ves 1. Learn techniques for represen3ng con$nuous-$me periodic signals using orthogonal sets of periodic basis func3ons. 2. Study proper3es of exponen$al, trigonometric and compact Fourier series, and condi3ons for their existence. 3. Learn the Fourier transform for non-periodic signal as an extension of Fourier series for periodic signals 4. Study the proper$es of the Fourier transform. Understand the concepts of energy and power spectral density.

2 4.1 Introduc5on How to deal with a complicated signal? 2

3 4.1 Introduc5on How to deal with a complicated signal? Using a series of sin(t), or cos(t) func3ons. Fourier Series and Fourier Transform x(t) = Asin(ωt +θ ) 3

4 4.1 Introduc5on How to deal with a complicated signal? Using a series of rectangular pulses Wavelet 4

5 4.1 Introduc5on What is Fourier, Fourier Series, and Fourier Transform?? Fourier is a man, a genius Name: Jean Bap3ste Joseph Fourier Year: Na5onality: French Fields: Mathema3cian, physicist, historian 5

6 4.1 Introduc5on Fourier Series and Fourier Transformer A weighted summa3on of Sines and Cosines of different frequencies can be used to represent periodic (Fourier Series), or non-periodic (Fourier Transform) func3ons. Is this true? People didn t believe that, including Lagrange, Laplace, Poisson, and other big wigs. But, yes, this is true, this is great!! 6

7 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons For a periodic signal for all t!x(t) which is periodic with period T 0 has the property!x(t +T) =!x(t) A T 0 /2 -A T 0 Periodic square-wave signal

8 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons Suppose that we wish to approximate this signal using just one trigonometric func3on 1.) Q: Should we use a sine or a cosine func3on? A: choose sine func3on. Because both and a sine func3on ( ) are odd symmetry. The fundamental period of sine func3on is T 0 as.!x (1) (t) = b 1 sin(ω 0 t)!x(t) b 1 sin(ωt)!x(t) 2.) Q: How should we adjust the parameters of the trigonometric func3on? A: find the parameters that can minimize the approxima$on error func3on.!ε 1 (t) =!x(t)!x (1) (t) =!x(t) b 1 sin(ω 0 t)

9 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons The best approxima3on to!x(t)!x (1) (t)!x(t) using only one trigonometric func3on is!x (1) (t) = b 1 sin(ω 0 t) ω 0 = 2π T 0!ε 1 (t) =!x(t)!x (1) (t) b 1 = 4A π A A -A T 0 -A

10 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons Let s try a two-frequency approxima3on to!x(t) error can be reduced.!x (2) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) and see if the approximate!ε 2 (t) =!x(t)!x (2) (t) =!x(t) b 1 sin(ω 0 t) b 2 sin(2ω 0 t) b 1 sin(ω 0 t) b 2 sin(2ω 0 t) b 1 = 4A π b 2 = 0 T 0

11 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons Let s try a three-frequency approxima3on to!x(t) error can be reduced.!x (3) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t)+ b 3 sin(3ω 0 t) b 1 = 4A π b 2 = 0 b 3 = 4A 3π and see if the approximate!ε 3 (t) =!x(t)!x (3) (t) =!x(t) b 1 sin(ω 0 t) b 2 sin(2ω 0 t) b 3 sin(3ω 0 t)!x(t) A!x (3) (t)!ε 3 (t) =!x(t)!x (3) (t) A -A T 0 -A

12 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons Let s try a 15-frequency approxima3on to error can be reduced.!x(t) and see if the approximate!x (15) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) b 15 sin(15ω 0 t)!ε 15 (t) =!x(t)!x (15) (t)!x(t)!x (15) (t)!ε 15 (t) =!x(t)!x (15) (t) A A -A T 0 -A

13 4.2.1 Approxima5ng a periodic signal with trigonometric func5ons We can draw some conclusion based on previous observa5ons: ² The normalized average power of the error signal!ε 3 (t) seems to be less than that of the error 1!ε (t) ² Consequently,!x (3) (t) is a be[er approxima3on to the signal than!x (1) (t) ² On the other hand, the peak value of the approximate error seems to be ±A for both 1!ε (t) and!ε 3 (t). If we try higher-order approxima3ons using more trigonometric basic func3on, the peak approxima3on error would s3ll be ±A.

14 4.2.2 Trigonometric Fourier Series (TFS) We may want to represent this signal using a linear combina3on of sinusoidal func3ons in the form: In a compact nota5on (trigonometric Fourier Series): Where!x(t) = a 0 + a 1 cos(ω 0 t) + a 2 cos(2ω 0 t) a k cos(kω 0 t) +... ω 0 = 2π f 0 +b 1 sin(ω 0 t) + b 2 sin(2ω 0 t) b k sin(kω 0 t) +...!x(t) = a 0 + a k cos(kw 0 t) + b k sin(kw 0 t) k=1 k=1 is the fundamental frequency in rad/s. and the sinusoidal func3ons with radian frequency ω 0,2ω 0,...kω 0 referred to as the basis func3ons ( cos(kω 0 t), sin(kω 0 t) ) are

15 4.2.2 Trigonometric Fourier Series (TFS)!x(t) = a 0 + a k cos(kw 0 t) + b k sin(kw 0 t) k=1 Synthesis equa5on: k=1 We need to determine the coefficients: a 0, a k, and b k

16 4.2.2 Trigonometric Fourier Series (TFS) Useful orthogonal sets: t 0 +T 0 cos(mω 0 t)cos(kω 0 t) dt = t 0 t 0 +T 0 sin(mω 0 t)sin(kω 0 t) dt = t 0 T 0 2, m = k 0, m k T 0 2, m = k 0, m k t 0 +T 0 cos(mω 0 t)sin(kω 0 t) dt = 0 t 0

17 4.2.2 Trigonometric Fourier Series (TFS) Synthesis equa5on:!x(t) = a 0 + a k cos(kw 0 t) + b k sin(kw 0 t) k=1 Analysis equa5on: k=1 a k = 2 t 0 +T 0!x(t)cos(kw 0 t)dt, for k =1,..., T t0 0 b k = 2 t 0 +T 0!x(t)sin(kw 0 t)dt, for k =1,..., T t0 0 a 0 = 1 T 0 t 0 +T 0 t 0!x(t)dt (dc component)

18 4.2.2 Trigonometric Fourier Series (TFS) Example 4.1 Trigonometric Fourier Series of a Periodic Pulse Train A pulse-train signal!x(t) with a period of T 0 = 3 seconds is shown as below. Determine the coefficients of the TFS representa3on of this signal.

19 4.2.2 Trigonometric Fourier Series (TFS) Example 4.4 Trigonometric Fourier Series of a Square Wave Determine the TFS for the periodic square wave shown as below.

20 4.2.3 Exponen5al Fourier Series (EFS) TFS!x(t) = a 0 + a k cos(kw 0 t) + b k sin(kw 0 t) k=1 k=1 cos(kω 0 t) = 1 ( 2 e jkw t 0 + e jkw t ) 0 sin(kω 0 t) = 1 2 j ( e jkw t 0 e jkw t ) 0 EFS!x(t) = k= c k e jkω 0 t

21 4.2.3 Exponen5al Fourier Series (EFS) General Format of Exponen5al Fourier Series!x(t) = c k e jkω t 0 k= EXAMPLES Single-tone Signals!x(t) = Acos(ω 0 t +θ) = A ( 2 e j(ω 0t+θ ) + e j(ω 0t+θ )) Comparing with general format = A ( 2 e jθ e jω 0t + e jθ e jω 0t ) c 1 = A 2 e jθ c 1 = A 2 e jθ

22 !x(t) = EFS Exponen5al Fourier Series (EFS) General Format of Exponen5al Fourier Series c k e jkω t 0!x(t) = a 0 + a k cos(kw 0 t) + b k sin(kw 0 t) k= k=1 TFS k=1 c k = 1 ( 2 a k jb k )

23 4.2.3 Exponen5al Fourier Series (EFS) General Format of Exponen5al Fourier Series!x(t) = k= c k e jkω 0 t We need to find out coefficient C k

24 4.2.3 Exponen5al Fourier Series (EFS) Useful orthogonal set t0 t 0 +T 0 e jmω 0t e jkω 0t dt = T 0, m = k 0, m k

25 4.2.3 Exponen5al Fourier Series (EFS) Synthesis equa5on:!x(t) = k= c k e jkω 0 t Analysis equa5on: c k = 1 T 0 t0 t 0 +T 0!x(t)e jkω 0t dt

26 4.2.3 Exponen5al Fourier Series (EFS) Example 4.5 Exponen3al Fourier Series For a Periodic Pulse Train Determine the EFS for the periodic square wave shown as below.!x(t) = k= c k = sin(kπ / 3) πk e j2πkt/3 sin(kπ / 3) πk c 0 =1/ 3

27 4.2.3 Exponen5al Fourier Series (EFS) Example 4.5 Exponen3al Fourier Series For a Periodic Pulse Train Determine the EFS for the periodic square wave shown as below.

28 4.2.3 Exponen5al Fourier Series (EFS) Example 4.10 Spectrum of half-wave rec3fied sinusoidal signal Determine the EFS coefficients and graph the line spectrum for the half-wave periodic signal!x(t) defined by. sin(w 0 t), 0 t T 0 / 2!x(t) = and!x(t +T 0 ) =!x(t) 0, T 0 / 2 t < T 0

29 4.2.5 Existence of Fourier Series Is it always possible to determine the Fourier series coefficients? Dirichlet Condi$on 3F ² Finite absolute value:!x(t) dt < T 0 0 ² Finite number of discon3nui3es in!x(t) ² Finite number of minima and maxima in one period

30 4.2.7 Proper5es of Fourier Series Linearity:!x(t) = c k e jkw 0t and!y(t) = d k e jkw 0t k= a 1!x(t)+ a 2!y(t) = k= [a 1 c k + a 2 d k ]e jkw 0t k= Where a 1 and a 2 are any two constants Time shiaing:!x(t) = k= c k e jkw 0t!x(t τ ) = k= [c k e jkw 0τ ]e jkw 0 t