EE104: Lecture 6 Outline
Announcements:
HW 1 due today, HW 2 posted
Review of Last Lecture
Additional comments on Fourier transforms
Review of time window Fourier transform
Useful Properties of Fourier Transforms
Key Fourier Transform Properties
Review of Last Lecture
Introduction to Fourier Transforms
Fourier Transform from Fourier Series
Fourier Transform Pairs
t
f
x(t)
X(f)=|X(f)|ejX(f)
For real signals, |X(f)=|X(-f)| and X(f)=-X(-f)
Additional FT Comments
Amplitude conveys information about signal’sfrequency content
Phase conveys little insight, except that a time delayresults in a linear phase shift
Dirichlet conditions are sufficient for FT to exist(not necessary)
Signals that are not absolutely integrable can have aFT (e.g. sin, cosine, constant, sinc)
Signals whose square is absolutely integrable have aFourier Transform (Plancerel’s thm: pg 23).
Rectangular Pulse Review
Rectangular pulse is a time window
FT is a sinc function, infinite frequency content
Shrinking time axis causes stretching of frequency axis
Signals cannot be both time-limited and bandwidth-limited
.5T
-.5T
A
t
f
Infinite Frequency Content
Useful Properties of FTs
Linearity
Represent signal as sum of others with known FT
Differentiation in Time
Useful for linear systems analysis (solving DEs)
Integration in Time
Simple multiplication in frequency
DC property
Integral of a signal determined by its FT at f=0.
Conjugation
Indicates symmetry of real signals
Parseval’s Relation
Power determined from time or frequency domains
Key Properties of FTs
Time scaling
Contracting in time yields expansion in frequency
Duality
Operations in time lead to dual operations in frequency
Fourier transform pairs are duals of each other
Frequency shifting
Multiplying in time by an exponential leads to afrequency shift.
Convolution and Multiplication
Multiplication in time leads to convolution in frequency
Convolution in time leads to multiplication in frequency
Main Points
Time delay leads to linear phase shifts in frequency
Signal power can be computed in time orfrequency domains
Time scaling contracts a signal along the time axis,which stretches it along the frequency axis
The time and frequency domains are duals of eachother in Fourier analysis.
Frequency shifting is obtained by modulating asignal in time.