Skip to content Skip to navigation

Connexions

You are here: Home » Content » DTFT Examples

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions 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.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

DTFT Examples

Module by: Don Johnson. E-mail the author

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: How to compute discrete-time Fourier transforms for decaying sequences.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Example 1

Let's compute the discrete-time Fourier transform of the exponentially decaying sequence sn=anun s n a n u n , where un u n is the unit-step sequence. Simply plugging the signal's expression into the Fourier transform formula,

Fourier Transform Formula

Sⅇⅈ2πf=∑n=-∞∞anunⅇ-ⅈ2πfn=∑n=0∞aⅇ-ⅈ2πfn S 2 f n a n u n 2 f n n 0 a 2 f n (1)

This sum is a special case of the geometric series.

Geometric Series

∀α,|α|<1:∑n=0∞αn=11−α α α 1 n 0 α n 1 1 α (2)
Thus, as long as |a|<1 a 1 , we have our Fourier transform.
Sⅇⅈ2πf=11−aⅇ-ⅈ2πf S 2 f 1 1 a 2 f (3)

Using Euler's relation, we can express the magnitude and phase of this spectrum.

|Sⅇⅈ2πf|=11−acos2πf2+a2sin22πf S 2 f 1 1 a 2 f 2 a 2 2 f 2 (4)
∠Sⅇⅈ2πf=-arctanasin2πf1−acos2πf S 2 f a 2 f 1 a 2 f (5)

No matter what value of aa we choose, the above formulae clearly demonstrate the periodic nature of the spectra of discrete-time signals. Figure 1 shows indeed that the spectrum is a periodic function. We need only consider the spectrum between -12 1 2 and 12 1 2 to unambiguously define it. When a>0 a 0 , we have a lowpass spectrum — the spectrum diminishes as frequency increases from 0 0 to 12 1 2 — with increasing a a leading to a greater low frequency content; for a<0 a 0 , we have a highpass spectrum (Figure 2).

Figure 1: The spectrum of the exponential signal ( a=0.5 a 0.5 ) is shown over the frequency range -22 -2 2 , clearly demonstrating the periodicity of all discrete-time spectra. The angle has units of degrees.
Figure 1 (spectrum10.png)
Figure 2: The spectra of several exponential signals are shown. What is the apparent relationship between the spectra for a=0.5 a 0.5 and a=-0.5 a -0.5 ?
Figure 2 (spectrum11.png)

Example 2

Analogous to the analog pulse signal, let's find the spectrum of the length- N N pulse sequence.

sn=1if0≤n≤N−10otherwise s n 1 0 n N 1 0 (6)

The Fourier transform of this sequence has the form of a truncated geometric series.

Sⅇⅈ2πf=∑n=0N−1ⅇ-ⅈ2πfn S 2 f n 0 N 1 2 f n (7)

For the so-called finite geometric series, we know that

Finite Geometric Series

∑n= n 0 N+ n 0 −1αn=α n 0 1−αN1−α n n 0 N n 0 1 α n α n 0 1 α N 1 α (8)
for all values of α α .

Exercise 1

Derive this formula for the finite geometric series sum. The "trick" is to consider the difference between the series'; sum and the sum of the series multiplied by α α .

Solution

α∑n= n 0 N+ n 0 −1αn−∑n= n 0 N+ n 0 −1αn=αN+ n 0 −α n 0 α n n 0 N n 0 1 α n n n 0 N n 0 1 α n α N n 0 α n 0 (9)
which, after manipulation, yields the geometric sum formula.

Applying this result yields (Figure 3.)

Sⅇⅈ2πf=1−ⅇ-ⅈ2πfN1−ⅇ-ⅈ2πf=ⅇ-ⅈπfN−1sinπfNsinπf S 2 f 1 2 f N 1 2 f f N 1 f N f (10)

The ratio of sine functions has the generic form of sinNxsinx N x x , which is known as the discrete-time sinc function, dsincx dsinc x . Thus, our transform can be concisely expressed as Sⅇⅈ2πf=ⅇ-ⅈπfN−1dsincπf S 2 f f N 1 dsinc f . The discrete-time pulse's spectrum contains many ripples, the number of which increase with N N , the pulse's duration.

Figure 3: The spectrum of a length-ten pulse is shown. Can you explain the rather complicated appearance of the phase?
Figure 3 (spectrum12.png)

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

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

| A lens (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions 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