1. EE104: Lecture 5 Outline Review of Last Lecture Introduction to Fourier Transforms Fourier Transform from Fourier Series Fourier Transform Pair and Signal Spectrum Example: Rectangular Pulse Time/Bandwidth Tradeoffs

2. Review of Last Lecture Basis Functions for Periodic Signals Fourier Series Transform Pair LTI Filtering of Exponentials Fourier Series Examples and Demo Fourier Series with Sinusoidal Bases Properties of Fourier Series h(t)

3. Properties of Fourier Series Linearity Multiplication Multiplication in time leads to convolution of FS Time Shifting Time shift leads to linear phase shift in FS Time Reversal Time reversal leads to index reversal Time Scaling Time scaling leads to frequency stretching Conjugation: Parseval’s Relation: Energy contained in FS

4. Introduction to Fourier Transforms The Fourier transform of a signal represents its spectral components. The Fourier transform and inverse provide a 1-1 mapping between time and frequency domains. x(t) t X(f) f

5. Fourier Series to Fourier Transform Repeat x(t) every T 0 seconds to get x p (t) Fourier series coefficients separated in frequency by f0=1/T0 As T 0 , samples in frequency domain become a continuous signal in f x(t) t X(f) f -T 0 0 T0T0 x p (t) 0 1/T 0 X p (f)

6. Fourier Transform Pair |X(f)| f  X(f) f Real signals have |X(f)|=-|X(f)| and  X(f)=-<X(-f)

7. Rectangular Pulse Rectangular pulse is a time window Fourier series coefficients of periodic square wave are weighted samples of X(f) 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

8. Main Points The Fourier transform represents the spectral components of a signal. The Fourier transform pair allows a signal to be uniquely represented in time or frequency domain Fourier transform usually represented in terms of amplitude and phase (symmetries for real signals) The Fourier transform of a rectangular pulse in time is a sinc function: critical transform pair A signal cannot be both time-limited and bandlimited