Sampling CT Signals: A Frequency Domain Perspective 2.1 2003/01/09 2003/03/05 Rob "The Kid" Nowak nowak@rice.edu Michael Haag mjhaag@rice.edu Rob "The Kid" Nowak nowak@rice.edu CT fourier transforms frequency FT nyquist sampling time The module will provide an introduction to sampling a signal in the frequency domain and go through a basic example.

Understanding Sampling in the Frequency Domain We want to relate x c t directly to x n . Compute the CTFT of x s t n x c n T δ t n T X s Ω t n x c n T δ t n T Ω t n x c n T t δ t n T Ω t n x n Ω n T n x n ω n X ω where ω Ω T and X ω is the DTFT of x n . X s Ω 1 T k X c Ω k Ω s X ω 1 T k X c Ω k Ω s 1 T k X c ω 2 k T where this last part is 2 -periodic.

Sampling

Speech Speech is intelligible if bandlimited by a CT lowpass filter to the band ±4 kHz. We can sample speech as slowly as _____?

Note that there is no mention of T or Ω s !

Relating x[n] to sampled x(t) Recall the following equality: x s t n n x n T δ t n T

Recall the CTFT relation: ↔ x α t 1 α X Ω α where α is a scaling of time and 1 α is a scaling in frequency. X s Ω X Ω T