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Chapter

Chapter

# The Fourier Transform of a Sequence

DOI link for The Fourier Transform of a Sequence

The Fourier Transform of a Sequence book

# The Fourier Transform of a Sequence

DOI link for The Fourier Transform of a Sequence

The Fourier Transform of a Sequence book

## ABSTRACT

In Chapter 6 we were able to treat a discrete-time signal as a formally continuous function by representing the sampled signal as a weighted impulse train
x
I
(
t
)
=
x
(
t
)
P
Δ
t
(
t
)
=
∑
ℓ
=
−
∞
∞
x
(
t
)
δ
(
t
−
ℓ
Δ
t
)
=
∑
ℓ
=
−
∞
∞
x
(
ℓ
Δ
t
)
δ
(
t
−
ℓ
Δ
t
)
,
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and we derived its Fourier transform in two forms (recall Theorem 6.11 and Corollary 6.13):
X
I
(
f
)
=
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{
x
I
(
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)
}
=
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=
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∞
∞
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Δ
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−
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2
π
f
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︸
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X
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=
−
∞
∞
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(
f
−
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Δ
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︸
F
{
x
(
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)
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*
F
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Δ
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(
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,
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where X(f) = ℱ{x(t)}. Note that X_{I}
(f +1/Δt) = X_{I}
(f), so X_{I}
(f) is a periodic function with period equal to the sampling rate ℝ = 1/Δt. Recall that if x(t) is band-limited with bandwidth F ≤ R, then we may extract X(f)= ℱ{x(t)} from the central period of X_{I}
(f); otherwise the shifted replicas of X (f) will overlap, and X_{I}
(f) = X (f) for f ∈ [−ℝ/2, ℝ/2].