ELEC212 Chapter 2 Discrete_Time_Signals_Systems

ELEC212 Chapter 2 Discrete_Time_Signals_Systems -...

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ELEC 212 Chapter 2: Discrete-Time Signals and Systems Prof. Matthew R. McKay 09-10 Fall ELEC 212 Discrete-Time Signals and Systems 2 Discrete-time signal and Linear System Theory How much do you remember? 09-10 Fall ELEC 212 Discrete-Time Signals and Systems 3 Discrete-Time Signals and Systems Representations How many signal and system representations do we have? You have learnt in ELEC 211 Signals and Systems: – Time-domain representation – Frequency-domain representation Anything else? () n j n j e n x e X ω −∞ = = ] [ ] [ n x ( ) j e X Forward Transform Inverse Transform Discrete-Time Fourier Transform: = π 2 0 ) ( 2 1 ) ( d e e X t x t j j In Engineering discipline, there are many transforms for discrete-time signals, such as •Discrete Cosine Transform (Application: Image Compression) •Z-transform (Application: Signal and System Analysis) •Hilbert Transform (Application: Communication System) •Fast Fourier Transform (Discrete Fourier Transform) (Applications: Communication System, Image Processing, Signal Processing) •Wavelet Transform (Applications: Image Compression, System Identification) 09-10 Fall ELEC 212 Discrete-Time Signals and Systems 4 Discrete-Time Signals and System Representations In this course, several important transforms will be discussed : Continuous-time Fourier Transform (CTFT) Discrete-time Fourier Transform (DTFT) Laplace Transform Z-transform Fast Fourier Transform (FFT) (A efficient algorithm for evaluating the elements in Discrete Fourier Transform (DFT) ) Let’s have a revision on Time-domain representation of discrete- time signals and systems first. n j n j e n x e X −∞ = = ] [ ] [ n x ( ) j e X Forward Transform Inverse Transform Discrete-Time Fourier Transform: = 2 0 ) ( 2 1 ) ( d e e X t x t j j
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09-10 Fall ELEC 212 Discrete-Time Signals and Systems 5 2.1 Basic Sequence and Sequence Operations • A sequence , in which n th number in the sequence is denoted as , is formally written as where n is an integer. x r n x [ n ] -2 -1 0 1 2 3 4 5 ]} [ { n x x = r ] [ n x 09-10 Fall ELEC 212 Discrete-Time Signals and Systems 6 Energy and Power of Sequence Signal Energy over finite interval: where is the magnitude Signal Energy over infinite interval: Signal Power over finite interval: Signal Power over infinite interval: In particular, if x [ n ] is a periodic signal with period N , – the signal power over an infinite interval is the same as the power over one period since the signal is repeated for every period N . = = 2 1 2 ] [ n n n n x E −∞ = = n n x E 2 ] [ = + = M M n M n x M P 2 ] [ 1 2 1 lim = + = 2 1 2 1 2 ] [ 1 1 n n n n x n n P = = 1 0 2 ] [ 1 N n n x N P 09-10 Fall ELEC 212 Discrete-Time Signals and Systems 7 Example Find the signal energy and power of the sequence shown below.
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ELEC212 Chapter 2 Discrete_Time_Signals_Systems -...

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