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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 ScienceThe 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:φ(F)= tan-1?(F)?(F)Real part How much of a cosine of that frequency you need Imaginary part How much of a sine of that frequency you need

Magnitude Amplitude of combined cosine and sine

Phase Relative proportions of sine and cosineThe Fourier Transform: Examples, Properties, Common Pairs

Example: Fourier Transform of a Cosine

f(t) =cos(2πst)F(u) =Z f(t)e-i2πutdt=Z cos(2πst)e-i2πutdt=Z cos(2πst) [cos(-2πut) +isin(-2πut)]dt=Z cos(2πst)cos(-2πut)dt+iZ cos(2πst)sin(-2πut)dt=Z cos(2πst)cos(2πut)dt-iZ cos(2πst)sin(2πut)dt0 except whenu=±s0 for allu=1 2

δ(u-s) +12

δ(u+s)The Fourier Transform: Examples, Properties, Common Pairs

Example: Fourier Transform of a Cosine

Spatial Domain Frequency Domain

cos(2πst)12

δ(u-s) +12

δ(u+s)0.20.40.60.81

-1 -0.5 0.5 1 -10-5510 0.2 0.4 0.6 0.8

1The Fourier Transform: Examples, Properties, Common Pairs

Sinusoids

Spatial Domain Frequency Domain

f(t)F(u)cos(2πst)12 [δ(u+s) +δ(u-s)] sin(2πst)12 i[δ(u+s)-δ(u-s)] The Fourier Transform: Examples, Properties, Common Pairs

Constant Functions

Spatial Domain Frequency Domain

f(t)F(u)1δ(u) a aδ(u)The Fourier Transform: Examples, Properties, Common Pairs

Delta Functions

Spatial Domain Frequency Domain

f(t)F(u)δ(t)1The Fourier Transform: Examples, Properties, Common Pairs

Square Pulse

Spatial Domain Frequency Domain

f(t)F(u)?

0 otherwisesinc(aπu) =sin(aπu)aπuThe Fourier Transform: Examples, Properties, Common Pairs

Square PulseThe Fourier Transform: Examples, Properties, Common Pairs

Triangle

Spatial Domain Frequency Domain

f(t)F(u)?

0 otherwisesinc2(aπu)The Fourier Transform: Examples, Properties, Common Pairs

Comb

Spatial Domain Frequency Domain

f(t)F(u)δ(tmodk)δ(umod 1/k) The Fourier Transform: Examples, Properties, Common Pairs

Gaussian

Spatial Domain Frequency Domain

f(t)F(u)e -πt2e-πu2The Fourier Transform: Examples, Properties, Common Pairs

Differentiation

Spatial Domain Frequency Domain

f(t)F(u)d dt

2πiuThe Fourier Transform: Examples, Properties, Common Pairs

Some Common Fourier Transform Pairs

Spatial DomainFrequency Domain

f(t)F(u)Cosinecos(2πst)Deltas1 2 [δ(u+s) +δ(u-s)]Sinesin(2πst)Deltas1 2 Combδ(tmodk)Combδ(umod 1/k)The Fourier Transform: Examples, Properties, Common Pairs

More Common Fourier Transform Pairs

Spatial DomainFrequency Domain

0 otherwiseSinc

2sinc

2(aπu)Gaussiane

-πt2Gaussiane -πu2Differentiationd dtRamp2πiuThe Fourier Transform: Examples, Properties, Common Pairs

Properties: Notation

LetFdenote the Fourier Transform:

F=F(f)

LetF-1denote 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) =aF(f)

The Fourier Transform: Examples, Properties, Common Pairs

Properties: Translation

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

2=f1(t-a)

F

1=F(f1)

F

2=F(f2)

then|F2|=|F1|

φ(F2) =φ(F1)-2πua

Intuition: magnitude tells you "how much", phase tells you "where".The Fourier Transform: Examples, Properties, Common Pairs

Change of Scale: Square Pulse RevisitedThe Fourier Transform: Examples, Properties, Common Pairs

Rayleigh"s Theorem

Total "energy" (sum of squares) is the same in either domain: |f(t)|2dt=? |F(u)|2duquotesdbs_dbs20.pdfusesText_26