Dienstag, 22. Februar 2011

EE5410 Discrete-time Fourier transform : aperiodic signals

EE5410 Discrete-time Fourier transform : aperiodic signals

how an aperiodic sequence can be thought of as a linear combination of complex exponentials.

1. In particular, the synthesis equation is in effect a representation of x[n] as a linear combination of complex exponentials infinitesimally close in frequency an with amplitudes X(e^jw)(dw/(2*PI) ).

2.For this reason, as in continuous time, the Fourier transform X(e^jw) will often be referred to as the spectrum of x[n], because it provides us with the information on how x[n] is composed of complex exponentials at different frequencies.

3. In particular, the Fourier coefficients ak of a periodic signal p[n] can be expressed in terms of EQUALLY spaced samples of the Fourier transform of a finite-duration, aperiodic signal x[t] that is equal to p[n] over ONE period and is zero otherwise.

4. discrete-time complex exponentials that differ in frequency by a multiple of 2*PI are identical.

5. For periodic discrete-time signals, the Fourier series coefficients are periodic and that the Fourier series representation is a finite SUM.

6. For aperiodic signals, X(e^jw) is periodic ( with period 2*PI) and that the synthesis equation involves an integration only over a frequency interval that produces distinct complex exponentials (ie., any interval of length 2*PI).

7. The periodicity of e^(jwn) as a function of w : w=0 and w=2*PI yield the same signal.

8. Signals at frequencies near these values or any other EVEN multiple of PI are slowly varying and therefore are all appropriately thought of as low-frequency signals.

9. The high frequencies in discrete time are the values of w near add multiples of PI.

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