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)|e j  X(f) For real signals, |X(f)=|X(-f)| and  X(f)=-  X(-f)

Additional FT Comments Amplitude conveys information about signal’s frequency content Phase conveys little insight, except that a time delay results in a linear phase shift Dirichlet conditions are sufficient for FT to exist (not necessary) Signals that are not absolutely integrable can have a FT (e.g. sin, cosine, constant, sinc) Signals whose square is absolutely integrable have a Fourier 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 a frequency 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 or frequency 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 each other in Fourier analysis. Frequency shifting is obtained by modulating a signal in time.