fourier transform properties n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Fourier Transform Properties PowerPoint Presentation
Download Presentation
Fourier Transform Properties

Loading in 2 Seconds...

play fullscreen
1 / 10

Fourier Transform Properties - PowerPoint PPT Presentation


  • 101 Views
  • Uploaded on

Fourier Transform Properties. t. F ( w ). w. - 6 p. - 4 p. - 2 p. 2 p. 4 p. 6 p. t. t. t. t. t. t. 0. Duality. Forward/inverse transforms are similar Example: rect(t/ t )  t sinc( w t / 2) Apply duality t sinc(t t /2)  2 p rect(- w / t )

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Fourier Transform Properties' - coby


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
duality

t

F(w)

w

-6p

-4p

-2p

2p

4p

6p

t

t

t

t

t

t

0

Duality
  • Forward/inverse transforms are similar
  • Example: rect(t/t)  t sinc(wt / 2)

Apply duality t sinc(t t/2)  2 p rect(-w/t)

rect(·) is even t sinc(t t /2)  2 p rect(w/t)

f(t)

1

t

-t/2

0

t/2

scaling
Scaling
  • Given and that a 0

|a| > 1: compress time axis, expand frequency axis

|a| < 1: expand time axis, compress frequency axis

  • Extent in time domain is inversely proportional to extent in frequency domain (a.k.a bandwidth)

f(t) is wider  spectrum is narrower

f(t) is narrower  spectrum is wider

shifting in time
Shifting in Time
  • Shift in time

Does not change magnitude of the Fourier transform

Shifts phase of Fourier transform by -wt0(so t0 is the slope of the linear phase)

  • Derivation

Let u = t – t0, so du = dt and integration limits stay same

sinusoidal amplitude modulation1

F(w)

1

w

-w1

w1

0

Y(w)

1/2 F(w+w0)

1/2 F(w-w0)

1/2

w

-w0 - w1

-w0 + w1

w0 - w1

w0 + w1

0

-w0

w0

Sinusoidal Amplitude Modulation
  • Example: y(t) = f(t) cos(w0 t)

f(t) is an ideal lowpass signal

Assume w1 << w0

  • Demodulation (i.e. recovery of f(t) from y(t)) is modulation followed by lowpass filtering
  • Similar derivation for modulation with sin(w0 t)
time differentiation property
Conditions

f(t)  0 when |t| 

f(t) is differentiable

Derivation of property:Given f(t) F(w)

Time Differentiation Property
summary
Summary
  • Definition of Fourier Transform
  • Two ways to find Fourier Transform

Use definition

Use transform pairs and properties