C Rectangular Function Pairs (5B) Continuous ime Rect Function Pairs
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Fourier ransform ypes Continuous ime Fourier Series CFS C n = 1 0 x t e j n 0 t dt x t = n=0 C n e j n 0t Continuous ime Fourier ransform CF X j = x t e j t d t x t = 1 X j e j t d C Rect (5B) 3
CFS and CF 0 = 1 0 C k 0 + 0 0 Period = 0 CFS (Continuous ime Fourier Series) C k = 1 0 sin(k / ) k / C 0 = 0 X ( j ω) = sin(ω /) ω/ + 4 π π + π + 4 π CF (Continuous ime Fourier ransform) X ( j ω) = / X ( j 0) = lim ω 0 + / e j ω t dt = cos(ω /) 1/ = sin(ω /) ω/ C Rect (5B) 4
CFS of a Periodic Pulse rain CFS of a Shifted Periodic Pulse rain CF of a Rectangular Pulse CF of a Shifted Rectangular Pulse C Rect (5B) 5
CFS Continuous ime Fourier Series CFS C n = 1 0 x t e j n 0 t dt x t = n=0 C n e j n 0t 0 = 1 0 C k 0 + 0 0 Period = 0 CFS (Continuous ime Fourier Series) C k = 1 0 sin(k / ) k / C 0 = 0 C Rect (5B) 6
Periodic Pulse rain CFS (1) Continuous ime Fourier Series C k = 1 0 x(t ) e j k t dt x(t) = n=0 C k e + j k t C k = 1 + 0 / x 0 / 0(t ) e j k ω t 0 dt 0 C k 0 = 0 / + 0 / e j k t dt [ = 1 + / e j k j k t] / C k 0 = 0 / + 0 / x 0 (t) e j k t dt = e j k / e + j k / j k C 0 = 1 0 sin (k /) k / Period = 0 0 C k 0 + 0 + k C Rect (5B) 7
Periodic Pulse rain CFS () Fundamental Frequency C k = 1 0 sin(k / ) k / = 1 0 sin( k /) k / = π 0 0 = π = π 0 = π 0 = 1 k = k π 0 k = k π sin( k ) Zeros at k = ±, ±4, = π 0 + 0 0 = 1 0 C k 0 = π 0 k C Rect (5B) 8
Periodic Pulse rain CFS (3) C k = 1 0 sin(k / ) k / = 1 0 sin( k /) k / Zeros at sin(k /) = 0 k / = ±n π k π 0 = ±nπ Zeros at k = ±n 0 Zeros at ω = k = ±n 0 the first zero = π 0 + 0 0 = 1 0 C k 0 0 = π 0 k C Rect (5B) 9
Periodic Pulse rain CFS (4) C k = 1 0 sin(k / ) k / = 0 sin(k /) k / = π 0 k = k π 0 C 0 = 0 k = ±n 0 Zeros at 0 C k the first zero 0 Period = 0 + 0 = 0 / 1 0 = π 0 0 = π 0 C Rect (5B) 10
CFS of a Periodic Pulse rain CFS of a Shifted Periodic Pulse rain CF of a Rectangular Pulse CF of a Shifted Rectangular Pulse C Rect (5B) 11
Shifted Square Wave CFS (1) Continuous ime Fourier Series C k = 1 0 x(t ) e j k t C k = 1 + 0 x 0 0(t ) e j k ω t 0 dt 0 C k 0 = 0 + 0 x 0 (t) e j k t dt dt x(t) = n=0 = π 0 C k e + j k t Fundamental Frequency 0 = π 0 = 1 = π = 0 + e j k t = e j k e 0 j k [ dt = 1 + e j k j k t]0 = 1 e j k ω0 j k = π 0 = 1 e j k π / 0 C k 0 = 1 e j k π / 0 C j k π/ k = 1 ( 1)k j k π/ 0 j π k 0 Period = 0 C 0 0 = 0 + e j 0 t dt = C 0 = 1 0 + + 0 C 4 0 C 3 1 j 3 π C 0 C 1 1 j π C 0 C 1 C C 3 C 4 1 1 1 j π 0 j 3 π 0 C Rect (5B) 1
Shifted Square Wave CFS () Period = 0 0 = 1 C k = 1 e j k π / 0 j k π C 0 = 0 0 + + 0 e + j k / Period = 0 0 + 0 0 = 1 C k = 1 0 sin(k / ) k / C 0 = 0 0 1 sin(k /) 0 k / = e + j k π / 0 ( = sin(k π / 0) k π = (e+ j k π / 0 e j k π / 0 ) j k π 1 e j k π / ) ( 0 = e + j k / 1 e j k π / ) 0 j k π j k π 1 0 C k C Rect (5B) 13
CFS of a Periodic Pulse rain CFS of a Shifted Periodic Pulse rain CF of a Rectangular Pulse CF of a Shifted Rectangular Pulse C Rect (5B) 14
CF Continuous ime Fourier ransform CF X j = x t e j t d t x t = 1 X j e j t d X ( j ω) = sin(ω / ) ω/ + 4 π π + π +4 π CF (Continuous ime Fourier ransform) X ( j ω) = / X ( j 0) = lim ω 0 + / e j ω t dt = cos(ω /) 1/ = sin(ω /) ω/ C Rect (5B) 15
Rect(t/) CF (1) Continuous ime Fourier ransform Aperiodic Continuous ime Signal X j = x t e j t d t x t = 1 X j e j t d X ( j ω) = / + / e j ω t dt [ = 1 + / j ω e j ωt] / = e j ω / e + j ω / j ω = sin(ω /) ω/ + X ( j 0) = lim ω 0 sin(ω /) ω/ = lim ω 0 cos(ω /) 1/ = sin(ω /) = 0 ω / = π n ω = π n ω = ± π, ± 4 π, ±6 π, 4 π π X ( j ω) = + π +4 π sin(ω / ) ω/ C Rect (5B) 16
Rect(t/) CF () Continuous ime Fourier ransform Aperiodic Continuous ime Signal X j = x t e j t d t x t = 1 X j e j t d X ( j ω) = / + / e j ω t dt = sin(ω /) ω/ X ( j ω) = π sin(ω / ) ω/ k π + 4 π π + π +4 π C Rect (5B) 17
CFS of a Periodic Pulse rain CFS of a Shifted Periodic Pulse rain CF of a Rectangular Pulse CF of a Shifted Rectangular Pulse C Rect (5B) 18
Shifted Rect(t/) CF (1) Continuous ime Fourier ransform Aperiodic Continuous ime Signal X j = x t e j t d t x t = 1 X j e j t d 0 + C Rect (5B) 19
Relation between CFS and CF C Rect (5B) 0
CF and CFS Continuous ime Fourier ransform X j = x t e j t d t x t = 1 X j e j t d X ( j ω) = sin (ω /) ω/ Aperiodic Continuous ime Signal + Continuous ime Fourier Series Periodic Continuous ime Signal C k = 1 0 x(t ) e j k t dt x(t) = C k e + j k t n=0 Period = 0 C k = 1 0 sin(k /) k / 0 + 0 + C Rect (5B) 1
CF CFS Aperiodic Continuous ime Signal Continuous ime Fourier ransform 0 x(t ) + As 0, x 0 (t) x(t ) Periodic Continuous ime Signal Continuous ime Fourier Series Period = 0 = π 0 0 0 + 0 x 0 (t) = + n= x(t n 0 ) + C Rect (5B)
CF and CFS as 0 (1) 0 = = 0 / sin(k /) k / C k 0 (k π/) sin(k 0 /4) k / 0 + 0 + 0 + 0 0 + 0 0 = 4 = 0 / 4 sin(k /) k / C k 0 (k π/4) sin(k 0 /8) k / 0 0 + 0 = 8 + 0 + 0 = 0 /8 sin(k /) k / 4 C k 0 4 (k π/8) sin(k 0 /16) k / 0 + + 0 8 8 = π 0 = π 8 0 = π C Rect (5B) 3
CF and CFS as 0 () 0 = 4 = 0 / 4 sin(k 0 /8) C k 0 k / 0 0 + + 0 + 0 4 +4 = π 0 = π 4 8 π 4 π π + π + 4 π + 8 π C k = 1 0 sin(k /) k / 1 0 0 π 0 π = 0 C Rect (5B) 4
CF of a Rect(t/) function (3) 0 = 4 1 0 4 +4 0 0 + + 0 + 0 8 π 4 π π + π + 4 π + 8 π 0 π π 0 0 = C k 0 = sin (k /) k / X ( j ω) = sin(ω /) ω/ X ( j ω) = lim k ω sin(k /) k / = sin(ω / ) ω/ C k = 1 0 sin(k /) k / X ( j ω) = sin(ω /) ω/ π k π + 4 π π + π +4 π C Rect (5B) 5
References [1] http://en.wikipedia.org/ [] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 003 [3] G. Beale, http://teal.gmu.edu/~gbeale/ece_0/fourier_series_0.html [4] C. Langton, http://www.complextoreal.com/chapters/fft1.pdf