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DCSP-11 Jianfeng Feng Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dsp.html

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1. Data transmission 2. Information theory

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3. Signal Representation idea: time vs. Frequency ( continuous FT)

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Sequences and their representation A sequence is an infinite series of real numbers { x(n) }, which is written { x(n) } = {…, x(-1),x(0),x(1),x(2), …,x(n), … }

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Sequences and their representation A sequence is an infinite series of real numbers { x(n) }, which is written { x(n) } = {…, x(-1),x(0),x(1),x(2), …,x(n), … } This can be used to represent a sampled signal, i.e. x(n) = x(nT), where x(t) is the original (continuous) function of time. (might prefer x[n]) Q: we can not work on x(t), how about x(n)?

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Discrete Time Fourier Transform (DTFT) The basic tool of signal analysis is the Fourier transform (DTFT)

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Definition and properties: The DTFT gives the frequency representation of a discrete time sequence with infinite length.

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Definition and properties: The DTFT gives the frequency representation of a discrete time sequence with infinite length. X( ): frequency domain x(n): time domain

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Find the DC term ? Calculate the power spectrum?

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the unit impulse

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Computation of the DTFT It is well known that the whole Fourier approach to signal analysis is based on the expansion of a signal in terms of sinusoids or, more precisely complex exponentials. In this approach we begin analyzing a signal by determining the frequencies contributing to its spectrum in terms of magnitudes and phases.

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For example, if a sequence is a sinusoid x(n)=cos( 0 n) of infinite length, its DTFT yields two `delta' functions,

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