4.3 Analysis of Non-periodic Con6nuous-Time Signals. We view this non-periodic signal as a periodic signal with period as infinite large.

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1 We view this non-periodic signal as a periodic signal with period as infinite large.

2 For periodic signal: c k = 1 T 0 T 0 /2 T 0 /2 x(t)e jkω 0t dt For non-periodic signal: we make T 0 -> c k T 0 = T 0 /2 T 0 /2 X(ω) = lim[c k T 0 ] = lim T 0 > T 0 > x(t)e jkω 0t dt T 0 /2 T 0 /2 = x(t)e jωt dt where w = kw 0 x(t)e jkω 0t dt

3 Shorthand nota<on: X(ω) = {x(t)} x(t) = 1 {X(w)} x(t) X(w)

4 x(t) : waveform in 6me domain X(f) : same waveform in frequency domain

5 How to understand Fourier transform? X( f ) = x(t)e j2π ft dt For X(f), at any frequency f, all the x(t) ( t from - to ) has contribu<on. x(t) = X( f )e j2π ft dt Similar For x(t), at any frequency t, all the X(f) ( f from - to ) has contribu<on.

6 Time Domain How to understand Fourier Series and Fourier Transform?!x (1) (t) = b 1 sin(ω 0 t)!x(t)!x (1) (t) Time t is from - to we call it <me domain b 1 = 4A π Only one frequency is used in this example: ω 0 = 2π T 0 or f 0 = 1/T 0

7 Time Domain How to understand Fourier Series and Fourier Transform?!x (2) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t)!x(t)!x (2) (t) Time t is from - to we call it <me domain b 1 = 4A π b 2 = 0 two frequencies are used in this example: ω 0 = 2π T 0 and 2ω 0

8 Time Domain How to understand Fourier Series and Fourier Transform?!x (3) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t)+ b 3 sin(3ω 0 t)!x(t)!x (3) (t) Time t is from - to we call it <me domain three frequencies are used in this example: ω 0 = 2π T 0 b 1 = 4A π b 2 = 0 b 3 = 4A 3π and 2ω 0 3ω 0

9 4.2.1 Approxima6ng a periodic signal with trigonometric func6ons Let s try a 15-frequency approxima<on to error can be reduced.!x(t) and see if the approximate!x (15) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) b 15 sin(15ω 0 t)!ε 15 (t) =!x(t)!x (15) (t)!x(t)!x (15) (t)!ε 15 (t) =!x(t)!x (15) (t) A A -A T 0 -A

10 Time Domain How to understand Fourier Series and Fourier Transform?!x (15) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) b 15 sin(15ω 0 t) Time t is from - to we call it <me domain 15 frequencies are used in this example: ω 0 = 2π T 0 b 1 = 4A π b 2 = 0 b 3 = 4A 3π b 4 = 0 b 5 = 4A 5π and 2ω 0 3ω 0,.

11 Time Domain b 4 in(4ω 0 t)!x (4) (t) b 3 in(3ω 0 t) b 2 sin(2ω 0 t) b 1 sin(ω 0 t)

12 Time Domain and Frequency Domain Frequency Domain

13 Time Domain and Frequency Domain sin(x) func<on can be viewed as a circle projected onto a line

14 Time Domain and Frequency Domain

15 6me domain x(t) frequency domain X(f) Example of Music

16 4.3.2 Existence of Fourier Transform s it always possible to determine the Fourier series coefficients? Dirichlet Condi;on 3F ² Finite absolute value: x(t) dt < ² Finite number of discon<nui<es in!x(t) ² Finite number of minima and maxima in one period

17 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse Using the forward Fourier transform integral, find the Fourier transform of the isolated rectangular pulse signal x(t) = A t τ

18 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse Frequency Domain

19 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse

20 4.3.5 Proper6es of Fourier Transform Linearity: x 1 (t) X 1 (w) and x 2 (t) X 2 (w) α 1 x 1 (t)+α 2 x 2 (t) α 1 X 1 (w)+α 2 X 2 (w) Where a 1 and a 2 are any two constants Duality: x(t) X(w) X(t) 2π x( w) x(t) X( f ) X(t) x( f )

21 4.3.5 Proper6es of Fourier Transform Symmetry of Fourier Transform: x(t): Real, m{x(t)} = 0 X * (w) = X( w) x(t): mage, Re{x(t)} = 0 X * (w) = X( w) Time ShiTing: x(t) X(w) x(t τ ) X(w)e jwτ Frequency ShiTing: x(t) X(w) x(t)e jw 0t X(w w 0 )

22 4.3.5 Proper6es of Fourier Transform Modula6on Property: x(t) X(w) x(t)cos(w 0 t) 1 2 X w w 0 ( ) + X ( w + w 0 ) Or x(t)cos(w 0 t) 1 2 X f f 0 ( ) + X ( f + f 0 ) x(t)sin(w 0 t) 1 2 X ( w w 0)e jπ /2 + X ( w + w 0 )e More general format x(t)cos(w 0 t +θ) 1 2 e jθ X f f 0 jπ /2 ( ) + e jθ X ( f + f 0 )

23 4.3.5 Proper6es of Fourier Transform Modula6on Property: Find the Fourier Transform of the modulated pulse given by cos(2π f 0 t), t < τ x(t) = 0, t < τ

24 4.3.5 Proper6es of Fourier Transform Modula6on Property:

25 4.3.5 Proper6es of Fourier Transform Convolu6on Property: x 1 (t) X 1 (w) x 2 (t) X 2 (w) x 1 (t)* x 2 (t) X 1 (w)x 2 (w) X 1 (w)* X 2 (w) x 1 (t)x 2 (t)