1.6.3 Spectral analysis via DFT
use DFT to approximate $ X(a) $ for a DT signal x(n)
Relationships of signals
<p>infinate duration signal: $ x(n) \Rightarrow DTFT \Rightarrow X(w) $
also infinate druation signal: $ x(n) \Rightarrow *truncation\Rightarrow \bar{x}(n) = x[n] $ finite duration
The periodic sample of $ x_p(n) = \sum_{l=-\infty}^{\infty} \bar{x}(n+lN) $
The DTFT of periodic sample is $ X(k) $
The DTFT of $ \bar{x}(n) \Rightarrow \bar{X}(w) \Rightarrow sampling\Rightarrow \bar{X}(k\frac{2\pi}{N}) = X(k) $
Example
$ x(n) = cos(w_0n) $
$ X(w) = rep_{2\pi}(\pi \delta(w-w_0) + \pi \delta(w+w_0)) $
consider $ \bar{x}(n) = x(n)w(n) $
- $ w(n) = 1 | 0 \le n \le N, 0 | else $
$ \bar{X}(w) = F(\bar{x}(n)) $
$ = \frac{1}{2\pi}X(w)*W(w) $
$ = \frac{1}{2}(W(w+w_0) + W(w-w_0)) $
Two sources of inaccuracies
- signal truncation $ \Rightarrow $ "leakage"
- frequency sampling $ \Rightarrow $ "picket fence effect"
--Drestes 15:05, 23 February 2009 (UTC)
