Intro Digital Signal Process
Intro Digital Signal Process ECE 714
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This 3 page Class Notes was uploaded by Miss Felicita Stiedemann on Thursday October 29, 2015. The Class Notes belongs to ECE 714 at University of New Hampshire taught by W. Miller in Fall. Since its upload, it has received 10 views. For similar materials see /class/231688/ece-714-university-of-new-hampshire in Computer Engineering at University of New Hampshire.
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Date Created: 10/29/15
Spectral Properties of Discrete Sequences The Discrete Time Fourier Transform The Discrete Time Fourier Transform oo Xw DTFTxn E Z xne39j nz oo Where Xw is the Fourier frequency spectrum ofthe discrete time signal xn e39jwn cosam j sinam Real Xw Z xn cosam n oo Imag Xw Z xn sinam n oo f physical time is given by t t n n 5 fs then the normalized frequency is defined as w2nfts2n where f is the physical frequency in Hz Note that the true Fourier spectrum as expressed in the DTFT can not in general be determined since the discrete sum is infinite in extent and the frequency variable on is continuous Thus quotspectral estimationquot is an important topic in DSP Periodicity ofthe Discrete Time Fourier Transform Since n is always an integer Z rmxm cosam and Z rm xn sinam are periodic in a with period a 2H for any signal xn Thus the Fourier spectrum Xw of any discrete time signal xn can only carry unique information within a normalized frequency bandwidth of Au 2n Since f w f52 n this corresponds to a physical frequency bandwidth of Af f5 Symmetry of the Discrete Time Fourier Transform Real X w Z xn cos am Z xn cosam RealXw Imag X w Z xn sin am Z xn sinam Imag Xw If the signal xn is real valued the real part of Xw has even symmetry centered at a 0 while the imaginary part of Xw has odd symmetry in centered at a 0 However this means that Xw in the interval 139 S a S 0 is totally determined by Xw in the interval 0 S a S n Combined with the periodicity described above this means that the Fourier spectrum Xw of any real valued discrete time signal xn can only carry unique information within a normalized frequency bandwidth of Au 1139 This corresponds to a physical frequency bandwidth of Af JCS2 This is one form of the well known Nyquist sampling criteria Convolution and Multiplication Discrete convolution xn wn E 2k xkwn k DTFTxn Xw DTFTwn Ww DTFTxn wn DTFT Z xkwn 10 kz oo DTFTxn wn Z lt Z xkwn kgt e39jwn nz oo k oo DTFT lel W00 Z lt Z xkwn kgt ej ke39j k nz oo k oo DTFTxn wn Z xke39j klt Z wn e39j lgt kz oo nlzn kz m DTFTxn wn XwWw The DTFT of the convolution oftwo sequences in the time domain is equal to the product in the frequency domain of the DTFT s of the two individual sequences It is less easy to show but DTFTxnwn Xw Ww The DTFT of the product oftwo sequences in the time domain is equal to the circular convolution in the frequency domain of the DTFT s of the two individual sequences LTI Systems xn gt LTI System gt yn 6n gt LTI System gt hn xn Z xk6n k k 3101 Z xkhn k 9601 M k and Yw XwHw where DTFThn Hw The Fourier spectrum ofthe output of an LT system is equal to the Fourier spectrum of the input times the Fourier spectrum of the impulse response of the system Thus Hw the DTFT of the impulse response ofa discrete time LT system expresses the effects of the LT system on the magnitude and phase of any input signal as a function of frequency and is called the frequency response ofthe system
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