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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 10 29 Copyright © Orchard Publications Summary and The correspondence between and is denoted as If is an point real discrete time function, only of the frequency components of are unique. The discrete time and frequency functions are defined as even or odd, in accordance with the relations The even and odd properties of the DFT are similar to those of the continuous Fourier trans- form and are listed in Table 10.2. The linearity property of the DFT states that The time shift property of the DFT states that The frequency shift property of the DFT states that The time convolution property of the DFT states that The frequency convolution property of the DFT states that X m [ ] x n ( ) W N mn n 0 = N 1 = x n [ ] 1 N --- X m ( ) W N m n n 0 = N 1 = x n [ ] X m [ ] x n [ ] X m [ ] x n [ ] N N 2 X m [ ] Even Time Function f N n [ ] f n [ ] = Odd Time Function f N n [ ] f n [ ] = Even Frequency Function F N m [ ] F m [ ] = Odd Frequency Function F N m [ ] F m [ ] = ax 1 n [ ] bx 2 n [ ] + + aX 1 m [ ] bX 2 m [ ] + + x n k [ ] W N km X m [ ] W N k m x n [ ] X m k [ ] x n [ ] h n [ ] X m [ ] H m [ ] x n [ ] y n [ ] 1 N --- X k [ ] Y m k [ ] k 0 = N 1 1 N --- X m [ ] Y m [ ] =

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Chapter 10 The DFT and the FFT Algorithm 10 30 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications The sampling theorem, also known as Shannon’s Sampling Theorem, states that if a continuous time function is band-limited with its highest frequency component less than , then can be completely recovered from its sampled values, if the sampling frequency if equal to or greater than . A typical digital signal processing system contains a low-pass analog filter, often called pre-sam- pling filter, to ensure that the highest frequency allowed into the system, will be equal or less the sampling rate so that the signal can be recovered. The highest frequency allowed by the pre- sampling filter is referred to as the Nyquist frequency, and it is denoted as . If a signal is not band-limited, or if the sampling rate is too low, the spectral components of the signal will overlap each another and this condition is called aliasing. If a discrete time signal has an infinite length, we can terminate the signal at a desired finite number of terms, by multiplying it by a window function. However, we must choose a suitable window function; otherwise, the sequence will be terminated abruptly producing the effect of leakage If in a discrete time signal some significant frequency component that lies between two spectral lines and goes undetected, the picket-fence effect is produced. This effect can be alleviated by
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