Chapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter

Transcription

1 Chapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter 1. Learn techniques for represen3ng discrete-)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 5.2 Fourier Series (EFS) Fourier Series Synthesis Discrete-Time Fourier Series Synthesis = k= c k e jkω 0 t!x[n] = k=0 j(2π /N )kn c k e Analysis equa@on: c k = 1 T 0 t0 t 0 +T 0!x(t)e jkω 0t dt Analysis equa@on: c k = 1 N n=0 j(2π /N )kn x[n]e

3 5.2.7 of Fourier Series Linearity Fourier Series x(t) c k Discrete-Time Fourier Series x[n] c k y(t) d k a 1 x(t)+ a 2 y(t) a 1 c k + a 2 d k y[n] d k a 1 x[n]+ a 2 y[n] a 1 c k + a 2 d k Where a 1 and a 2 are any two constants

4 5.2.7 of Fourier Series Time shil Fourier Series!x(t) = k=!x(t τ ) = c k e jkw 0t k= [c k e jkw 0τ ]e jkw 0 t Discrete-Time Fourier Series!x[n] =!x[n m] = k=0 k=0 c k e j(2π /N )kn c k e j(2π /N )kn j(2π /N )km e

5 5.3 Analysis of Non-periodic Signals Discrete-Time Fourier Transform Synthesis (inverse): Analysis (forward): x[n] = 1 π X(Ω)e jωn dω 2π π X(Ω) = n= x[n]e jωn 2πk 2M +1 > Ω

6 5.3 Analysis of Non-periodic Signals Fourier Transform Synthesis (inverse): x(t) = 1 2π X(ω)e jwt dw Discrete-Time Fourier Transform Synthesis equa@on (inverse): x[n] = 1 π X(Ω)e jωn dω 2π π Analysis equa@on (forward): Analysis equa@on (forward): X(ω) = x(t)e jwt dt X(Ω) = n= x[n]e jωn

7 5.3.2 Existence of Fourier Transform s it always possible to determine the Fourier series coefficients? ² Absolute summable: x[n] < n= ² Square - summable: x[n] 2 < n=

8 5.3.5 of Fourier Transform Linearity: x 1 [n] X 1 (Ω) and x 2 [n] X 2 (Ω) α 1 x 1 [n]+α 2 x 2 [n] α 1 X 1 (Ω)+α 2 X 2 (Ω) Periodicity: Where a 1 and a 2 are any two constants X(Ω+ 2πr) = X(Ω) for all integers r

9 5.3.5 of Fourier Transform Time ShiLing: x[n] X(Ω) x[n m] X(Ω)e jωm Frequency ShiLing: x[n] X(Ω) x[n]e jω 0n X(Ω Ω 0 )

10 5.3.5 of Fourier Transform Property: x 1 [n] X 1 (Ω) x 2 [n] X 2 (Ω) x 1 [n]* x 2 [n] X 1 (Ω)X 2 (Ω) X 1 (Ω)* X 2 (Ω) x 1 [n]x 2 [n]

11 5.4 Energy and Power in Frequency Domain Parseval s Theorem: For a periodic power signal x(t) Con@nue-Time 1 t x(t) T 0 dt = c k T t0 0 k= 1 N Discrete-Time x[n] 2 2 = c k k=0 k=0 For a non-periodic power signal Con@nue-Time Discrete-Time x(t) 2 dt = X( f ) 2 df x[n] 2 = 1 2π k=0 π π X(Ω) 2 dω

12 5.4 Energy and Power in Frequency Domain Power Spectral Density: S x (Ω) = 2π c k 2 δ(ω kω0 ) k=

13 5.4 Energy and Power in Frequency Domain For a energy signal x(t) the autocorrela@on func@on is r xx [m] = x[n]x[n + m] n=

14 5.5 System Concept System (frequency response) mpulse response (h[n]) Fourier Transform System func3on (H(Ω)) H(Ω) = h[n] { } = h[n]e jωn n= n general, H(w) is a complex func3on of w, and can be wrijen in polar form as: H(Ω) = H(Ω) e jθ(ω)

15 5.8 Discrete Fourier Transform DTFS DTFT Synthesis (inverse): DFT x[n] = k=0 n = 0, 1,., N-1 j(2π /N )kn c k e x[n] = 1 2π π X(Ω)e jωn dω Analysis equa@on (forward): π x[n] = 1 N k=0 j(2π /N )kn X[k]e n = 0, 1,., N-1 c k = 1 N n=0 j(2π /N )kn x[n]e X(Ω) = n= x[n]e jωn X[k] = n=0 j(2π /N )kn x[n]e k = 0, 1,., N-1 k = 0, 1,., N-1

16 5.8 Discrete Fourier Transform of the DFT to the DTFT DTFT DFT X(Ω) = n= x[n]e jωn X[k] = n=0 j(2π /N )kn x[n]e The DFT of a length-n signal is equal to its DTFT evaluated at a set of N angular frequencies equally spaced in the interval [0, 2π). Let an indexed set of angular frequencies be defined as Ω k = 2πk, k = 0,1,..., N 1 N X[k] = X(Ω) = n=0 j(2π /N )kn x[n]e

17 Why do we need DFT? 5.8 Discrete Fourier Transform ² The signal x[n] and its DFT X[k] each have N samples, making the discrete Fourier transform prac3cal for computer implementa3on. ² Fast and efficient algorithm, know as fast Fourier transforms (FFTs), are available for the computa3on of the DFT. ² DFT can be used for approxima3ng other forms of Fourier series and transforms for both con3nuous-3me and discrete-3me system. ² Dedicated processors are available for fast and efficient. computa3on of the DFT with minimal or no programming needed.