Skip to content Skip to navigation

Connexions

You are here: Home » Content » Examing Reconstruction Relations

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Examing Reconstruction Relations

Module by: Justin Romberg. E-mail the author

Summary: This module focus on examing reconstruction relations.

Introduction

We've seen that if f⁢t f t is bandlimited to −π π , 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 f s n is just a periodic version of the CTFT of f⁢t f t

Let's examine the situation in general

Figure 1
Figure 1 (fig2.png)

Relationship of CTFT and DTFT

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

Figure 2
Figure 2 (fig3.png)

We will use the following notation. Note that we will use different variables, ΩΩ and ωω, so that we do not get confused. f⁢t CTFT F⁢i⁢Ω f t CTFT F Ω f⁢n⁢T= f s ⁢n CTFT F s ⁢ei⁢ω f n T f s n CTFT F s ω f⁢t=12⁢π⁢∫−∞∞F⁢i⁢Ω⁢ei⁢ω⁢nd n f t 1 2 n F Ω ω n

F s ⁢ei⁢ω=∑n=−∞∞ f s ⁢n⁢e−(i⁢ω⁢n)=∑n=−∞∞f⁢n⁢T⁢e−(i⁢ω⁢n)=∑n=−∞∞12⁢π⁢∫−∞∞F⁢i⁢Ω⁢ei⁢Ω⁢n⁢TdΩ⁢e−(i⁢ω⁢n)=12⁢π⁢∫−∞∞F⁢i⁢Ω⁢∑n=−∞∞ei⁢n⁢(Ω⁢T−ω)dΩ=12⁢π⁢∫−∞∞F⁢i⁢Ω⁢2⁢π⁢∑r=−∞∞δ⁢ω−(Ω⁢T−2⁢π⁢r)dΩ=∫−∞∞F⁢i⁢Ω⁢∑r=−∞∞δ⁢ω−(Ω⁢T−2⁢π⁢r)dΩ=∫−∞∞F⁢i⁢Ω⁢∑r=−∞∞δ⁢(−T)⁢(Ω−ω−2⁢π⁢rT)dΩ 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 (1)
Recall, property of δ⁢t δ t
δ⁢a⁢Ω=1|a|⁢δ⁢Ω δ a Ω 1 a δ Ω (2)
F s ⁢ei⁢ω=∫−∞∞F⁢i⁢Ω⁢∑r=−∞∞1T⁢δ⁢Ω−ω−2⁢π⁢rTdΩ=1T⁢∑r=−∞∞∫−∞∞F⁢i⁢Ω⁢δ⁢Ω−ω−2⁢π⁢rTdΩ=1T⁢∑r=−∞∞F⁢i⁢ω−2⁢π⁢rT 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 (3)
F s ⁢ei⁢ω F s ω is a sum of shifted (and scaled) F⁢i⁢ω F ω 's. In order to get to our final results in Equation 3, we used the sifting property.

Figure 3: Illustration of sampling f⁢t f t with a period of TT.
Figure 3 (exrec_f1.png)

Fs ⁢ei⁢ω=1T⁢∑r=,∞∞F⁢i⁢ω−2⁢π⁢rT 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⁢i⁢Ω→F⁢i⁢ωT → F Ω F ω T

Figure 4
Figure 4 (exrec_f2.png)

note:

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

Periodic repetition

Fs ⁢ei⁢ω=1T⁢∑r=−∞∞F⁢i⁢ω−2⁢π⁢rT Fs ω 1 T r F ω 2 r T What this equation says is that we "plop down" a copy of 1T⁢F⁢i⁢ωT 1 T F ω T at every multiple of 2⁢π 2 (and sum up)..

Figure 5: You can see the periodic nature of Fs ⁢ei⁢ω Fs ω . Note that F⁢i⁢ωT F ω T is zero for everything beyond the interval of −π π .
(exrec_f3.png)

note:

Fs ⁢ei⁢ω Fs ω is 2⁢π 2 -periodic, just like we know it should be.
Recall, T=sample period T sample period (i.e. each sample in the discrete-time has a distance of TT between them). Before, we saw that if T=1 T 1 (sample on the integers) and f⁢t f t is bandlimited to −π π we can reconstruct f⁢t f t exactly from its samples fs ⁢n fs n .

General Case

Say f⁢t f t is bandlimited to −ΩB ΩB ΩB ΩB . Then if we sample with period T≤π ΩB T ΩB , we can reconstruct f⁢t f t exactly from its samples fs ⁢n fs n . Show this following the exact same steps as in the −π π , T=1 T 1 case.

  • If f⁢t f t is bandlimited to −ΩB ΩB ΩB ΩB , then F⁢i⁢ωT F ω T , T≤π ΩB T ΩB is zero outside of −π π

Figure 6
(a) The CTFT of signal f⁢t ft
Figure 6(a) (mrec_f1.png)
(b) Signal is zero outside of −π π
Figure 6(b) (mrec_f2.png)

  • then 1T⁢∑r=,∞∞F⁢i⁢ω−2⁢π⁢rT 1 T r F ω 2 r T = DTFT of fs ⁢n=f⁢n⁢T fs n f n T is just a periodic version of 1T⁢F⁢i⁢ωT 1 T F ω T Fs ⁢ei⁢ω=1T⁢∑r=−∞∞F⁢i⁢ω−2⁢π⁢rT Fs ω 1 T r F ω 2 r T

Figure 7: DTFT of Sampled Signal
Figure 7 (mrec_f3.png)

  • turning fs ⁢n fs n into an impulse sequence fimp ⁢t=∑n=−∞∞ fs ⁢n⁢δ⁢t−n⁢T fimp t n fs n δ t n T

Figure 8
Figure 8 (mrec_f4.png)

  • From here we make some calculations and we get fimp ⁢t↔∫−∞∞ fimp ⁢t⁢e−(i⁢ω⁢t)dt=∫−∞∞∑n=−∞∞ fs ⁢n⁢δ⁢t−n⁢T⁢e−(i⁢ω⁢t)dt ↔ fimp t t fimp t ω t t n fs n δ t n T ω t so, fimp ⁢t=∑n=−∞∞ fs ⁢n⁢∫−∞∞δ⁢t−n⁢T⁢e−(i⁢ω⁢t)dt=∑n=−∞∞ fs ⁢n⁢e−(i⁢ω⁢T⁢n)= Fs ⁢ei⁢(ω⁢T) fimp t n fs n t δ t n T ω t n fs n ω T n Fs ω T

Figure 9
Figure 9 (mrec_f3.png)
Figure 10: CTFT of fimp ⁢i⁢ω fimp ω is just a stretched out version of DTFT of fs ⁢n fs n
Figure 10 (mrec_f5.png)

  • to recover f ~ ⁢t=f⁢t f ~ t f t from fimp ⁢t fimp t , we lowpass filter with cutoff frequency πT T (also the filter has a gain of 1T 1 T ).

Figure 11
(a) CTFT of fimp ⁢i⁢ω fimp ω
Figure 11(a) (mrec_f5.png)
(b) Lowpass filter
Figure 11(b) (mrec_f6.png)
(c) Reconstructed signal, which is identical to the original
Figure 11(c) (mrec_f7.png)

If G⁢i⁢ω G ω is "ideal", then F ~ ⁢i⁢ω F ~ ω and the signal is reconstructed perfectly.

Figure 12: Block diagram briefly illustrating the reconstruction process.
Figure 12 (recon_blk3.png)

Example 1: You try one ...

Starting with the following signal, f⁢t ft, and its CTFT, F⁢i⁢ω 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.

Figure 13: Sample and reconstruct the above signal, f⁢t ft.
Figure 13 (mrec_eg1.png)

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks