Introduction We've seen that if f t is bandlimited to , we can reconstruct it from its samples at the integers exactly. The key to this was the fact that the DTFT of f s n is just a periodic version of the CTFT of f t Let's examine the situation in general

Relationship of CTFT and DTFT What is the relationship of the CTFT of f t to the DTFT of f s n f t sampled at n T ?

We will use the following notation. Note that we will use different variables, Ω and ω, so that we do not get confused. f t  CTFT F Ω f n T f s n  CTFT F s ω f t 1 2 n F Ω ω n F s ω n f s n ω n n f n T ω n n 1 2 Ω F Ω Ω n T ω n 1 2 Ω F Ω n n Ω T ω 1 2 Ω F Ω 2 r δ ω Ω T 2 r Ω F Ω r δ ω Ω T 2 r Ω F Ω r δ T Ω ω 2 r T Recall, property of δ t δ a Ω 1 a δ Ω F s ω Ω F Ω r 1 T δ Ω ω 2 r T 1 T r Ω F Ω δ Ω ω 2 r T 1 T r F ω 2 r T F s ω is a sum of shifted (and scaled) F ω 's. In order to get to our final results in , we used the sifting property.

Illustration of sampling f t with a period of T. Fs ω 1 T r F ω 2 r T

Relationship Let us take a minute to examine this relation in 2 steps:

Axis Scaling → F Ω F ω T

→ Ω T ω , this covers the r 0 case in the sum above.

Periodic repetition Fs ω 1 T r F ω 2 r T What this equation says is that we "plop down" a copy of 1 T F ω T at every multiple of 2 (and sum up)..

You can see the periodic nature of Fs ω . Note that F ω T is zero for everything beyond the interval of . Fs ω is 2 -periodic, just like we know it should be. Recall, T sample period (i.e. each sample in the discrete-time has a distance of T between them). Before, we saw that if T 1 (sample on the integers) and f t is bandlimited to we can reconstruct f t exactly from its samples fs n .

General Case Say f t is bandlimited to ΩB ΩB . Then if we sample with period T ΩB , we can reconstruct f t exactly from its samples fs n . Show this following the exact same steps as in the , T 1 case. If f t is bandlimited to ΩB ΩB , then F ω T , T ΩB is zero outside of

The CTFT of signal ft Signal is zero outside of then 1 T r F ω 2 r T = DTFT of fs n f n T is just a periodic version of 1 T F ω T Fs ω 1 T r F ω 2 r T

DTFT of Sampled Signal turning fs n into an impulse sequence fimp t n fs n δ t n T

From here we make some calculations and we get ↔ fimp t t fimp t ω t t n fs n δ t n T ω t so, fimp t n fs n t δ t n T ω t n fs n ω T n Fs ω T

CTFT of fimp ω is just a stretched out version of DTFT of fs n to recover f ~ t f t from fimp t , we lowpass filter with cutoff frequency T (also the filter has a gain of 1 T ).

CTFT of fimp ω Lowpass filter Reconstructed signal, which is identical to the original If G ω is "ideal", then F ~ ω and the signal is reconstructed perfectly.

Block diagram briefly illustrating the reconstruction process. You try one ... Starting with the following signal, ft , and its CTFT, F ω , use the same steps above to sample the signal, convert it into an impulse train, and then filter it to recreate the original signal.

Sample and reconstruct the above signal, ft .