Magnitude and Phase The Fourier Transform: Examples ...

The Fourier Transform: Examples, Properties, Common Pairs

The Fourier Transform: Examples, Properties, Common Pairs

CS 450: Introduction to Digital Signal and Image Processing Bryan Morse

BYU Computer Science

The Fourier Transform: Examples, Properties, Common Pairs

Magnitude and Phase

Remember: complex numbers can be thought of as (real,imaginary) or (magnitude,phase).

Magnitude:

|F | =

(F )2 + (F )2 1/2

Phase:

(F )

=

tan-1

(F ) (F )

Real part Imaginary part

Magnitude Phase

How much of a cosine of that frequency you need How much of a sine of that frequency you need

Amplitude of combined cosine and sine Relative proportions of sine and cosine

The Fourier Transform: Examples, Properties, Common Pairs

The Fourier Transform: Examples, Properties, Common Pairs

Example: Fourier Transform of a Cosine

Z

f (t) = cos(2st)

F (u) =

f (t ) e-i2ut dt

Z-

=

cos(2st ) e-i2ut dt

Z-

=

cos(2st) [cos(-2ut) + i sin(-2ut)] dt

Z-

Z

=

cos(2st) cos(-2ut) dt + i cos(2st) sin(-2ut) dt

Z-

Z

-

=

cos(2st) cos(2ut) dt - i cos(2st) sin(2ut) dt

-

-

0 except when u = ?s

0 for all u

=

1 2

(u

-

s)

+

1 2

(u

+

s)

The Fourier Transform: Examples, Properties, Common Pairs

Example: Fourier Transform of a Cosine

Spatial Domain

Frequency Domain

1 0.5

-0.5 -1

cos(2st )

0.2 0.4 0.6 0.8

1

1 2

(u

-

s)

+

1 2

(u

+

s)

1

0.8

0.6

0.4

0.2

-10

-5

5

10

The Fourier Transform: Examples, Properties, Common Pairs

Odd and Even Functions

Sinusoids

Even

f (-t) = f (t) Symmetric

Cosines Transform is real

Odd

f (-t) = -f (t) Anti-symmetric

Sines Transform is imaginary

for real-valued signals

Spatial Domain f (t)

Frequency Domain F (u)

cos(2st ) sin(2st )

1 2

[(u

+

s)

+

(u

-

s)]

1 2

i

[(u

+

s)

-

(u

-

s)]

The Fourier Transform: Examples, Properties, Common Pairs

Constant Functions

Spatial Domain Frequency Domain

f (t)

F (u)

1

(u)

a

a (u)

The Fourier Transform: Examples, Properties, Common Pairs

Delta Functions

Spatial Domain Frequency Domain

f (t)

F (u)

(t )

1

The Fourier Transform: Examples, Properties, Common Pairs

Square Pulse

Spatial Domain f (t)

Frequency Domain F (u)

1 if -a/2 t a/2 0 otherwise

sinc(au)

=

sin(au) au

The Fourier Transform: Examples, Properties, Common Pairs

Square Pulse

The Fourier Transform: Examples, Properties, Common Pairs

Triangle

Spatial Domain f (t)

1 - |t| if -a t a 0 otherwise

Frequency Domain F (u)

sinc2(au)

The Fourier Transform: Examples, Properties, Common Pairs

Comb

Spatial Domain Frequency Domain

f (t)

F (u)

(t mod k)

(u mod 1/k )

The Fourier Transform: Examples, Properties, Common Pairs

Gaussian

Spatial Domain Frequency Domain

f (t)

F (u)

e-t 2

e-u2

The Fourier Transform: Examples, Properties, Common Pairs

Differentiation

Spatial Domain Frequency Domain

f (t)

F (u)

d dt

2iu

The Fourier Transform: Examples, Properties, Common Pairs

Some Common Fourier Transform Pairs

Spatial Domain f (t)

Cosine Sine Unit Constant Delta Comb

cos(2st ) sin(2st )

1 a (t ) (t mod k)

Frequency Domain

F (u)

Deltas Deltas Delta

1 2

[(u

+

s)

+

(u

-

s)]

1 2

i

[(u

+

s)

-

(u

-

s)]

(u)

Delta

a(u)

Unit

1

Comb

(u mod 1/k )

The Fourier Transform: Examples, Properties, Common Pairs

More Common Fourier Transform Pairs

Spatial Domain f (t)

Square

Triangle Gaussian Differentiation

1 if -a/2 t a/2

0 otherwise

1 - |t| if -a t a

0

otherwise

e-t 2

d dt

Frequency Domain F (u)

Sinc

sinc(au)

Sinc2

Gaussian Ramp

sinc2(au)

e-u2 2iu

The Fourier Transform: Examples, Properties, Common Pairs

Properties: Notation

Let F denote the Fourier Transform: F = F(f )

Let F -1 denote the Inverse Fourier Transform: f = F -1(F )

The Fourier Transform: Examples, Properties, Common Pairs

Properties: Linearity

Adding two functions together adds their Fourier Transforms together: F(f + g) = F(f ) + F(g)

Multiplying a function by a scalar constant multiplies its Fourier Transform by the same constant:

F(af ) = a F(f )

The Fourier Transform: Examples, Properties, Common Pairs

Properties: Translation

Translating a function leaves the magnitude unchanged and adds a constant to the phase. If

f2 = f1(t - a) F1 = F (f1) F2 = F (f2)

then

|F2| = |F1| (F2) = (F1) - 2ua

Intuition: magnitude tells you "how much", phase tells you "where".

The Fourier Transform: Examples, Properties, Common Pairs

Change of Scale: Square Pulse Revisited

The Fourier Transform: Examples, Properties, Common Pairs

Rayleigh's Theorem

Total "energy" (sum of squares) is the same in either domain:

|f (t)|2 dt =

|F (u)|2 du

-

-

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