Q2. if r(t) has the Fourier transform X(w) = jw+b, find V(w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21

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Q2. if r(t) has the Fourier transform X(w) = jw+b, find V (w) for the following:
a) v(t) = x(5t - 4)
b) v(t) = t²x(t)
c) v(t) = x(t)ei21
Transcribed Image Text:Q2. if r(t) has the Fourier transform X(w) = jw+b, find V (w) for the following: a) v(t) = x(5t - 4) b) v(t) = t²x(t) c) v(t) = x(t)ei21
TABLE 3.1 Properties of the Fourier Transform
Property
Linearity
Right or left shift in time
Time scaling
Time reversal
Multiplication by a power of t
Multiplication by a complex exponential
Multiplication by sin (wot)
Multiplication by cos(wot)
Differentiation in the time domain
Integration in the time domain
Convolution in the time domain
Multiplication in the time domain
Parseval's theorem
Special case of Parseval's theorem
Duality
Transform Pair/Property
ax(t) + bv(t)→aX(w) + bV(w)
x(tc)X(w)e-jose
x (at) < > ² x ( =) a >
X
a
x(−1)→X(-w) = X(w)
a>0
d"
t"x(t) → j" don X (w) n = 1, 2, ...
dw"
x(t)ejat →X(w - wo) wo real
x(t) sin(wt) ↔ { [X(w + w) − X(w − w)]
x(t) cos(wit) → [X(w + wo) + X(w − w)]
din X(t) → (jw)"X(w) n = 1, 2,...
1
[x(x) dx → — X (w) + 7X (0)8(w)
jw
x(t) *v(t) →X(w)V(w)
x(t)v(t) → __X(w)*V(w)
[x(1)v(1) dt =
1
[~_* x²(1) dt = 2 /
2TT
X(t)→2πx(-w)
[XV (oo) das
1X (w) ² do
-xx
TABLE 3.2 Common Fourier Transform Pairs
1, -∞ < t < ∞ → 2π8 (W)
−0.5 + u(t) ↔ -
jw
1
jw
u(t) <> πδ(ω) +
8(t) →1
8(tc)e-jwc, c any real number
1
jw + b²
e-btu(t).
ejut → 2πd (w - wo), wo any real number
τω
2π
P:(t)→7 sinc-
b>0
7 sinc→2πp, (w)
27
2|t|
τω
(1 - 2!!!) p.,(1)→ sinc²(e)
T
sinc²(# )+2π(1 - 2!!)P, (c)
2
T
cos(wot) →T[8(w + wo) + d(w = wo)]
cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)]
sin(wot) → jπ[8(w + wo) − 8(w - wo)]
sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
Transcribed Image Text:TABLE 3.1 Properties of the Fourier Transform Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power of t Multiplication by a complex exponential Multiplication by sin (wot) Multiplication by cos(wot) Differentiation in the time domain Integration in the time domain Convolution in the time domain Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality Transform Pair/Property ax(t) + bv(t)→aX(w) + bV(w) x(tc)X(w)e-jose x (at) < > ² x ( =) a > X a x(−1)→X(-w) = X(w) a>0 d" t"x(t) → j" don X (w) n = 1, 2, ... dw" x(t)ejat →X(w - wo) wo real x(t) sin(wt) ↔ { [X(w + w) − X(w − w)] x(t) cos(wit) → [X(w + wo) + X(w − w)] din X(t) → (jw)"X(w) n = 1, 2,... 1 [x(x) dx → — X (w) + 7X (0)8(w) jw x(t) *v(t) →X(w)V(w) x(t)v(t) → __X(w)*V(w) [x(1)v(1) dt = 1 [~_* x²(1) dt = 2 / 2TT X(t)→2πx(-w) [XV (oo) das 1X (w) ² do -xx TABLE 3.2 Common Fourier Transform Pairs 1, -∞ < t < ∞ → 2π8 (W) −0.5 + u(t) ↔ - jw 1 jw u(t) <> πδ(ω) + 8(t) →1 8(tc)e-jwc, c any real number 1 jw + b² e-btu(t). ejut → 2πd (w - wo), wo any real number τω 2π P:(t)→7 sinc- b>0 7 sinc→2πp, (w) 27 2|t| τω (1 - 2!!!) p.,(1)→ sinc²(e) T sinc²(# )+2π(1 - 2!!)P, (c) 2 T cos(wot) →T[8(w + wo) + d(w = wo)] cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)] sin(wot) → jπ[8(w + wo) − 8(w - wo)] sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
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