DiscreteFS_06

# DiscreteFS_06 - Discrete-Time Fourier Series It was shown...

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Discrete-Time Fourier Series It was shown that for an LTI, discrete-time system which is stable or satisfies the dominance condition that an exponential input produces an exponential output, that is, ( ) nn zH z z An input signal which reproduces itself at the system output is said to be an eigenfunction of the system, and the constant ( H ( z ) in this case) is its eigenvalue. The above result applies for any complex quantity z . Let ( ) 0 for 0, 1, 2, jk n k ne k ϕ == ± ± " These are all eigenfunctions of LTI systems. These functions are all special cases in which . 0 jk ze = Now consider a periodic, discrete-time signal having a fundamental period N . Then ( ) ( ) xn N xn += It can be shown that such a periodic signal can be expanded in a Fourier series, having a finite-number of terms () 0 1 0 0 2 where N jk n k k xn Xe N π = =Ω = In this expansion all of the exponential signals are periodic with period N . This is easily shown 0 00 0 0 0 2 2 jk N jk n N jk n jk N jk n jk n jk n jk N k k nN e e e e n ϕϕ Ω+ ΩΩ = = = = = The Fourier series defined above for discrete-time signals is called the Discrete-Time Fourier Series (DTFS). Though the DTFS is similar to the Fourier Series for continuous-time signals, it differs in a couple of important ways.

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2 1. The series contains a finite number of terms equal to the period N of the periodic signal. In contrast, the time-continuous Fourier series contains an infinite number of terms. 2. The quantity S 0 appearing in the complex exponential plays the same role in the DTFS as the fundamental frequency T 0 plays in the continuous-time Fourier series. In the DTFS the product is an angle measured in radians as is the product 0 n for the continuous-time series. Because t is measured in seconds, is a 0 t ω 0 frequency measured in radians per second. In contrast, the integer variable n has no units and so is an angle measured in radians. 0 It is customary to write the DTFS in a somewhat different form. Let 0 2 j j N N We e π = ± Using this notation the DTFS can be written () 1 0 N N nk k k xn XW = = Now write out the series for different values of n from 1 to N- 1. ( ) 01 2 N - 1 12 1 2 N - 1 21 24 2 N - 1 31 36 2 N - 1 1 2 N - 1 0 +X 1 + +X 2 +X 3 +X 1+ + X NN N N N N N N N N N xX X X X WX W W X W W X W W xN X XW W =+ + + + −= + + " " " " ## " ( ) 11 On the left side of this set of equations are the known values of the periodic signal. On the right side are the Fourier coefficients and the values of raised to different powers. N W But and its powers are all known and are just different constants. So the right side N W
3 contains the unknown Fourier coefficients multiplied by different constants. This then is a set of N linear equations in N unknowns. In principle, these equations can be solved to find the values of the Fourier coefficients.

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DiscreteFS_06 - Discrete-Time Fourier Series It was shown...

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