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

2. 1. Data transmission 2. Information theory

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

4. 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), … }

5. 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)?

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

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

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

10. Find the DC term ? Calculate the power spectrum?

11. the unit impulse

12. 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.

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