FOURIER TRANSFORM OF SIGNALS AND SYSTEMS TSKS01 Digital - PowerPoint PPT Presentation

TSKS01 DIGITAL COMMUNICATION Repetition and Examples FOURIER TRANSFORM OF SIGNALS AND SYSTEMS TSKS01 Digital Communication - Repetition

LTI Systems Definition: A system that is linear and time-invariant is referred to as a linear time-invariant (LTI) system . #(") ℎ " & " Definition: The convolution of the signals !(#) and %(#) is denoted by (! ∗ %)(#) and is defined as * ! ∗ % # = ( ! + % # − + -+ . )* The convolution is a commutative operation: ! ∗ % # = % ∗ ! # . TSKS01 Digital Communication - Repetition

Output of an LTI Systems Theorem: Let !(#) be the input to an energy-free LTI system with impulse response ℎ(#) , then the output of the system is & # = ! ∗ ℎ # . Proof: Let * +(#) denote the output for an arbitrary input +(#) , then . & # = * !(#) = * , ! / 0 # − / 2/ -. Linear . , ! / * 0 # − / 2/ = -. Time−inv . , ! / ℎ # − / 2/ = ! ∗ ℎ # = -. TSKS01 Digital Communication - Repetition

Frequency Domain *(+) = sinc + = sin 0+ ! " = $ " + 1 − $(" − 1) 0+ 1 −1 1 " + Frequency domain representation Time-domain signal Image from: https://en.wikipedia.org/wiki/Fourier_transform TSKS01 Digital Communication - Repetition

Fourier Transform Fourier transform + % ' , *-./01 2' ! " = ℱ %(') = ∫ Fourier transform: *+ + |% ' |2' < ∞ Exists if ∫ *+ + ! " , -./01 2" ℱ *6 !(") = ∫ Inverse transform: *+ Common terminology | ! " | Amplitude spectrum: arg !(") Phase spectrum: TSKS01 Digital Communication - Repetition

Fourier Transform – Examples ) * = cos 2/0 Complex exponential: ! " = $ %&'( 1 " + 3 sin(2/0 1 ") 8 0 = 9(0 − 0 1 ) ! " = cos 2/0 1 " Cosine: 8 0 = 1 1 + 1 2 9 0 − 0 2 9(0 + 0 1 ) ! " = sin 2/0 1 " Sine: 8 0 = 1 1 − 1 32 9 0 − 0 32 9(0 + 0 1 ) ! " = < " + 1 − <(" − 1) Rectangle pulse: 8 0 = sinc 0 = sin /0 /0 TSKS01 Digital Communication - Repetition

Fourier Transform – Properties Let ! " = ℱ %(') and ) " = ℱ *(') Convolution à Product: ℱ (% ∗ *)(') = ! " )(") Product à Convolution: ℱ % ' *(') = (! ∗ ))(") Time shift à Phase shift: ℱ %(' − -) = !("). /01234 TSKS01 Digital Communication - Repetition

+(%) Properties – Example !(#) −1 1 # % * % = + % , !"#$% & # = !(# − 1) % 0 2 # Image from: https://en.wikipedia.org/wiki/Fourier_transform TSKS01 Digital Communication - Repetition

Example – Baseband to Passband ℱ " # = %(') Recall: = 1 . + 1 ℱ cos 2-' . # 2 0 ' − ' 2 0(' + ' . ) Consequence: . #) = 1 . + 1 ℱ " # cos(2-' 2 % ' − ' 2 %(' + ' . ) Baseband: $(!) Passband: 3 . + 3 4 $ ! − ! 4 $(! + ! . ) −" − ! ! . + " . ! ! −" " −! ! 0 . . TSKS01 Digital Communication - Repetition

Frequency Response of LTI System Theorem: Let !(#) be the input to an energy-free LTI system with impulse response ℎ(#) , then the output of the system is & # = ! ∗ ℎ # . Definition: *(+) = ℱ ℎ(#) is called the frequency response. / |ℎ # |1# < ∞ Only exists for stable systems: ∫ ./ Property: The input and output of LTI systems are related as '(") ℎ " ) " = (' ∗ ℎ)(") ((%) #(%) , % = ( % #(%) TSKS01 Digital Communication - Repetition