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### lecture5

Course: EE 104, Fall 2009
School: Stanford
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Word Count: 393

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Lecture EE104: 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 Review of Last Lecture x p (t ) = n=- c e n j 2nf 0 t cn = 1 x p (t )e - j 2nf 0 t dt T0 T0 Basis Functions for Periodic Signals Fourier Series Transform Pair LTI Filtering of...

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Lecture EE104: 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 Review of Last Lecture x p (t ) = n=- c e n j 2nf 0 t cn = 1 x p (t )e - j 2nf 0 t dt T0 T0 Basis Functions for Periodic Signals Fourier Series Transform Pair LTI Filtering of Exponentials e j 2ft Fourier Series Examples and Demo Fourier Series with Sinusoidal Bases Properties of Fourier Series h(t) H ( f )e j 2ft Properties of Fourier Series Linearity Multiplication Time Shifting Multiplication in time leads to convolution of FS Time shift leads to linear phase shift in FS Time reversal leads to index reversal Time scaling leads to frequency stretching Time Reversal Time Scaling * x * (t ) {c- n } Conjugation: p Parseval's Relation: Energy contained in FS 1 2 | x p (t ) | dt = | cn |2 T0 T0 n=- 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) X(f) t f Fourier to Series Fourier Transform x(t) -T0 0 T0 X(f) t 0 f Repeat x(t) every T0 seconds to get xp(t) Fourier series coefficients separated in frequency by f0=1/T0 1/T As T0 , samples in frequency domain become a continuous signal in f Fourier Transform Pair X( f ) = x (t ) = |X(f)| - x (t )e - j 2nft dt X ( f )e j 2ft df X(f) - f f Real signals have |X(f)|=-|X(f)| and < X(f)=-<X(-f) Rectangular Pulse A -.5T .5T Infinite Frequency Content t f x (t ) = Arect (t / T ) X ( f ) = ATsinc ( fT ) 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 canno...

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