CTFT of Rectangular Pulse Functions (3B) PDF

Lecture 11 The Fourier transform

the Fourier transform of a signal f is the function. F(ω) = ∫. ∞. −∞ f(t)e shifted rectangular pulse: f(t) = {. 1 1 − T ≤ t ≤ 1 + T. 0 t < 1 − T or t > ...

The Fourier Transform

shifting a function f(t) by t0. The easy way to ... is Example 2 where we saw that the Fourier transform of the rectangular pulse rect(t) of height one and.

Properties of the Fourier Transform

g(t) is a pulse of width 2 and can be obtained by shifting the symmetrical rectangular pulse p1(t) = { 1. −1 ≤ t ≤ 1. 0 otherwise by 4 units to the right

Example: the Fourier Transform of a rectangle function: rect(t)

Consider the Fourier coefficients. Let's define a function F(m) that incorporates both cosine and sine series coefficients with the sine series distinguished

Fourier Transform Rectangular Pulse Example : rectangular pulse

-2. -1. 0. 1. 2. 1. -2. -1. 0. 1. 2. 1. +. = )(. )( )( 4. 2 tptpty. +. = Page 3. 9. Example : Time-Shift. ( ). │. ⎠. ⎞. │. ⎝. ⎛. = ↔. │. ⎩. │. ⎨. ⎧.

HELM 24: Fourier Transforms

The Fourier transform of a Rectangular Pulse. If pa(t) = { 1. −a<t<a. 0 Use the time-shifting property to find the Fourier transform of the function g(t) ...

EE 261 - The Fourier Transform and its Applications

function that's as good as ψ. Now suppose xT = 0 meaning that. 〈xT

Chapter 4: Frequency Domain and Fourier Transforms

One might guess that the. Fourier transform of a sinc function in the time domain is a rect function in Fourier transform is that it gets shifted by the same ...

Lecture 11 - More Fourier Transform.pdf

Feb 20 2011 ◇ Find the Fourier transform of the gate pulse x(t) given by: ◇ This pulse is rect(t/τ) dleayed by 3τ/4 sec. ◇ Use time-shifting theorem

CTFT of Rectangular Pulse Functions (3B)

5 Aug 2013 Spectrum Plots of the CTFT of a Shifted Rectangular Pulse. CTFT of Rectangular Pulse Functions (3B) ... Continuous Time Fourier Transform.

Fourier Transform.pdf

Determine the Fourier transform of a rectangular pulse shown in the following figure Therefore the amplitude spectrum of the time shifted signal is the.

Lecture 11 The Fourier transform

the Fourier transform of a signal f is the function shifted rectangular pulse: f(t) = {. 1 1 ? T ? t ? 1 + T. 0 t < 1 ? T or t > 1 + T.

Chapter 4: Frequency Domain and Fourier Transforms

One might guess that the. Fourier transform of a sinc function in the time domain is a rect function in frequency domain. This turns out to be correct as could

DTFT of Periodic Pulse Functions (3B)

5 Aug 2013 ... the DTFT of a Rectangular Pulse. ? Spectrum Plots of the DTFT of a Shifted Rectangular Pulse ... DTFT (Discrete Time Fourier Transform).

Fourier Transform Rectangular Pulse Example : rectangular pulse

Fourier Transform. 1. 2. Rectangular Pulse. T dt e. T c t j. 1. 1. 1. 5.0. 5.0. 0. 0. 0 Example : Time-Shift. ( ). ?. ?. ?. ?. ?. ?. = ?. ?. ?.

Discrete-Time Fourier Transform

7-1 DTFT: Fourier Transform for Discrete-Time Signals Another common signal is the L-point rectangular pulse which is a finite-length time.

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Figure 4: Shifting property of the ?-function. and then by the Shifting Theorem equation 26

A Tables of Fourier Series and Transform Properties

with period T = 2?/?. Fourier series Ck. Time shifting Table A.2 Properties of the continuous-time Fourier transform ... Even rectangular pulse.

Lecture 8 ELE 301: Signals and Systems

Linearity Theorem: The Fourier transform is linear; that is given two rect(t). Linearity Exam. Cuff (Lecture 7). ELE 301: Signals and Systems.

Young Won Lim

8/5/13CTFT of a Rectangular PulseCTFT of a Shifted Rectangular Pulse Spectrum Plots of the CTFT of a Rectangular PulseSpectrum Plots of the CTFT of a Shifted Rectangular PulseCTFT of Rectangular Pulse Functions (3B)

Young Won Lim

8/5/13 Copyright (c) 2009 - 2013 Young W. Lim.

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CT.3B Pulse CTFTCTFT of a Rectangular PulseCTFT of a Shifted Rectangular Pulse Spectrum Plots of the CTFT of a Rectangular PulseSpectrum Plots of the CTFT of a Shifted Rectangular Pulse

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CT.3B Pulse CTFTCTFT of a Rectangular Pulse (1)T

-T 2+T 2

Continuous Time Fourier TransformCTFT

X(jω)=sin(ωT/2)

ω/2

X(j0)=T

X(jω)=sin(ωT/2)

ω/2

-4π T T +2π

T+4π

T-2π

T =T⋅sinc(fT)

X(jω)=∫-∞

x(t)e-jωtdtx(t)=1

2π∫-∞

X(jω)e+jωtdω

1 x(t)=rect(2t T)

5Young Won Lim

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CT.3B Pulse CTFTCTFT of a Rectangular Pulse (2)

Continuous Time Fourier TransformAperiodic Continuous Time SignalX(jω)=∫-T/2 +T/2 e-jωtdt =[-1 jωe-jωt ]-T/2 +T/2 =-e-jωT/2-e+jωT/2 jω =sin(ωT/2)

ω/2

X(j0)=limω→0

sin(ωT/2)

ω/2

=limω→0 T 2 cos(ωT/2) 1/2=T T -T 2+T 2

X(jω)=sin(ωT/2)

ω/2

-4π T T +2π

T+4π

T-2π

X(jω)=∫-∞

x(t)e-jωtdtx(t)=1

2π∫-∞

X(jω)e+jωtdω

6Young Won Lim

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CT.3B Pulse CTFTZero Crossings of (1)

X(jω)=∫-∞

x(t)e-jωtdt

ω=2π

X(jk2π

T)=0ω=k2π

Tsinc(fT)

X(jω)=sin(ωT/2)

ω/2=T⋅sinc(fT)Zeros at

1 x(t) e-jωt= cos(ωt) +jsin(ωt) x(t)cosωtdt=0 x(t)sinωtdt=0

7Young Won Lim

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CT.3B Pulse CTFTZero Crossings of (2) T

X(jω)=∫-∞

x(t)e-jωtdt

ω=4π

Tsinc(fT)

X(jk2π

T)=0ω=k2π

TX(jω)=sin(ωT/2)

ω/2=T⋅sinc(fT)Zeros at

1 x(t) e-jωt= cos(ωt) +jsin(ωt) x(t)cosωtdt=0 x(t)sinωtdt=0

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CT.3B Pulse CTFTt=±1,±2,⋯Zeros atNormalized Sinc functionfT=±1,±2,⋯Zeros at

ω=±2π

T,±4π

T,⋯

sinc(t)=sin(πt) πt

X(jω)=sin(ωT/2)

ω/2=Tsin(ωT/2)

ωT/2=Tsin(πfT)

πfT

X(f)=T⋅sinc(fT)

sinc(x)=sin(x) xUnnormalized Sinc function x=±π,±2π,⋯Zeros atωT/2=±π,±2π,⋯Zeros at

X(jω)=T⋅sinc(ωT/2)

f=±1

T,±2

T,⋯Zero Crossings of (3)

Tsinc(fT)

9Young Won Lim

8/5/13

CT.3B Pulse CTFTt=±1,±2,⋯Zeros atNormalized Sinc functionfT=±1,±2,⋯Zeros at

ω=±2π

T,±4π

T,⋯

sinc(t)=sin(πt) πt

X(jω)=sin(ωT/2)

ω/2=Tsin(ωT/2)

ωT/2=Tsin(πfT)

πfT

X(f)=T⋅sinc(fT)

f=±1

T,±2

T,⋯

X(jω)=sin(ωT/2)

ω/2

-4π T T +2π

T+4π

T-2π

TZero Crossings of (4)

Tsinc(fT)

1 -T 2+T 2

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CT.3B Pulse CTFTSummary : CTFS of a Rectangular Pulse+2π TContinuous Time Fourier TransformAperiodic Continuous Time Signal

X(jω)=∫-T/2

+T/2 e-jωtdt +4π

T-2π

T-4π

T Tquotesdbs_dbs4.pdfusesText_8

[PDF] CTFT of Rectangular Pulse Functions (3B)

Young Won Lim

8/5/13CTFT of a Rectangular PulseCTFT of a Shifted Rectangular Pulse Spectrum Plots of the CTFT of a Rectangular PulseSpectrum Plots of the CTFT of a Shifted Rectangular PulseCTFT of Rectangular Pulse Functions (3B)

Young Won Lim

8/5/13 Copyright (c) 2009 - 2013 Young W. Lim.

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8/5/13

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CT.3B Pulse CTFTCTFT of a Rectangular Pulse (1)T

Continuous Time Fourier TransformCTFT

X(jω)=sin(ωT/2)

ω/2

X(j0)=T

X(jω)=sin(ωT/2)

ω/2

T+4π

T-2π

X(jω)=∫-∞

2π∫-∞

X(jω)e+jωtdω

5Young Won Lim

8/5/13

CT.3B Pulse CTFTCTFT of a Rectangular Pulse (2)

ω/2

X(j0)=limω→0

ω/2

X(jω)=sin(ωT/2)

ω/2

T+4π

T-2π

X(jω)=∫-∞

2π∫-∞

X(jω)e+jωtdω

6Young Won Lim

8/5/13

CT.3B Pulse CTFTZero Crossings of (1)

X(jω)=∫-∞

ω=2π

ω=2π

X(jk2π

T)=0ω=k2π

Tsinc(fT)

X(jω)=sin(ωT/2)

ω/2=T⋅sinc(fT)Zeros at

7Young Won Lim

8/5/13

CT.3B Pulse CTFTZero Crossings of (2) T

X(jω)=∫-∞

ω=4π

ω=4π

Tsinc(fT)

X(jk2π

T)=0ω=k2π

TX(jω)=sin(ωT/2)

ω/2=T⋅sinc(fT)Zeros at

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8/5/13

ω=±2π

T,±4π

T,⋯

X(jω)=sin(ωT/2)

ω/2=Tsin(ωT/2)

ωT/2=Tsin(πfT)

πfT

X(f)=T⋅sinc(fT)

X(jω)=T⋅sinc(ωT/2)

T,±2

T,⋯Zero Crossings of (3)

Tsinc(fT)

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ω=±2π

T,±4π

T,⋯

X(jω)=sin(ωT/2)

ω/2=Tsin(ωT/2)

ωT/2=Tsin(πfT)

πfT

X(f)=T⋅sinc(fT)