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Erik Strand
pit
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3e38b018
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3e38b018
authored
6 years ago
by
Erik Strand
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Add Fourier series notes
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---
title
:
Fourier Series
---
## Definition
Let $$f:
\m
athbb{R}
\r
ightarrow
\m
athbb{C}$$ be a reasonably well behaved $$L$$-periodic function
(i.e. $$f(x) = f(x + L)$$ for all $$x$$). Then $$f$$ can be expressed as an infinite series
$$
f(x) =
\s
um_{n
\i
n
\m
athbb{Z}} A_n e^{2
\p
i i n x / L}
$$
The values of the coefficients $$A_n$$ can be found using the orthogonality of basic exponential
sinusoids. For any $$m
\i
n
\m
athbb{Z}$$,
$$
\b
egin{align
*
}
\i
nt_{-L/2}^{L/2} f(x) e^{-2
\p
i i m x / L} dx
&=
\i
nt_{-L/2}^{L/2}
\s
um_{n
\i
n
\m
athbb{Z}} A_n e^{2
\p
i i n x / L} e^{-2
\p
i i m x / L} dx
\\
&=
\s
um_{n
\i
n
\m
athbb{Z}} A_n
\i
nt_{-L/2}^{L/2} e^{2
\p
i i (n - m) x / L} dx
\\
&=
\s
um_{n
\i
n
\m
athbb{Z}} A_n L
\d
elta_{nm}
\\
&= L A_m
\e
nd{align
*
}
$$
where the second to last line uses
[
Kronecker delta
](
https://en.wikipedia.org/wiki/Kronecker_delta
)
notation. Thus by the periodicity of $$f$$ and the exponential function
$$
\b
egin{align
*
}
A_n &=
\f
rac{1}{L}
\i
nt_{-L/2}^{L/2} f(x) e^{-2
\p
i i n x / L} dx
\\
&=
\f
rac{1}{L}
\i
nt_{0}^{L} f(x) e^{-2
\p
i i n x / L} dx
\e
nd{align
*
}
$$
## Relation to the Fourier Transform
Now suppose we have a function $$f:
\m
athbb{R}
\r
ightarrow
\m
athbb{C}$$, and we construct an
$$L$$-periodic version
$$
f_L(x) =
\s
um_{n
\i
n
\m
athbb{Z}} f(x + nL)
$$
The coefficients of the Fourier Series of $$f_L$$ are
$$
\b
egin{align
*
}
L A_n &=
\i
nt_0^L
\s
um_{m
\i
n
\m
athbb{Z}} f(x + mL) e^{-2
\p
i i n x / L} dx
\\
&=
\s
um_{m
\i
n
\m
athbb{Z}}
\i
nt_0^L f(x + mL) e^{-2
\p
i i n x / L} dx
\\
&=
\s
um_{m
\i
n
\m
athbb{Z}}
\i
nt_{mL}^{(m + 1)L} f(x) e^{-2
\p
i i n (x - mL) / L} dx
\\
&= e^{-2
\p
i i n m}
\s
um_{m
\i
n
\m
athbb{Z}}
\i
nt_{mL}^{(m + 1)L} f(x) e^{-2
\p
i i x n / L} dx
\\
&=
\i
nt_
\m
athbb{R} f(x) e^{-2
\p
i i x n / L} dx
\\
&=
\h
at{f}(n/L)
\e
nd{align
*
}
$$
so
$$
f_L(x) =
\f
rac{1}{L}
\s
um_{n
\i
n
\m
athbb{Z}}
\h
at{f}(n / L) e^{2
\p
i i n x / L}
$$
Thus the periodic summation of $$f$$ is completely determined by discrete samples of $$
\h
at{f}$$.
This is remarkable in that an uncountable set of numbers (all the values taken by $$f_L$$ over one
period) can be determined by a countable one (the samples of $$
\h
at{f}$$). Even more incredible, if
$$f$$ has finite bandwidth then only a finite number of the samples will be nonzero. So the
uncountable set of numbers is determined by a finite one.
## Derivation of the Discrete Time Fourier Transform
We can apply the result above to $$
\h
at{f}$$ as well. Recall that the Fourier transform of
$$
\h
at{f}$$ is $$f(-x)$$. Thus the periodic summation of the Fourier transform of $$f$$ is
$$
\b
egin{align
*
}
\h
at{f}_L(x) &=
\f
rac{1}{L}
\s
um_{n
\i
n
\m
athbb{Z}} f(-n / L) e^{2
\p
i i n x / L}
\\
&=
\f
rac{1}{L}
\s
um_{n
\i
n
\m
athbb{Z}} f(n / L) e^{-2
\p
i i n x / L}
\e
nd{align
*
}
$$
This is precisely the definition of the discrete time Fourier transform.
If $$f$$ is time-limited, then we'll have only a finite number of nonzero samples. But then
$$
\h
at{f}$$ is necessarily not bandwidth limited, so the tails of $$
\h
at{f}$$ will overlap in the
periodic summation. On the other hand, if $$f$$ is bandwidth limited, for sufficiently large $$L$$
we can recover $$
\h
at{f}$$. To do so perfectly requires an infinite number of samples, but in
practice reasonably bandwidth limited signals can still be recovered quite well from a finite number
of samples.
## Poisson Resummation
By taking $$x = 0$$ we see that
$$
\s
um_{n
\i
n
\m
athbb{Z}} f(nL) =
\f
rac{1}{L}
\s
um_{n
\i
n
\m
athbb{Z}}
\h
at{f}(n / L)
$$
This is known as the
[
Poisson summation formula
](
https://en.wikipedia.org/wiki/Poisson_summation_formula
)
.
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...
...
@@ -2,7 +2,7 @@
title
:
Fourier Transforms
---
To really make sense of chapter 2 I needed to review the properties of Fourier
T
ransforms. These
To really make sense of chapter 2 I needed to review the properties of Fourier
t
ransforms. These
notes are based on my prior knowledge and some helpful websites:
-
[
Properties of Fourier Transform
](
http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
)
...
...
@@ -177,7 +177,7 @@ for spectral power analysis).
## Transforms of Gaussians
A Fourier
T
ransform that comes up frequently is that of a Gaussian. It can be calculated by
A Fourier
t
ransform that comes up frequently is that of a Gaussian. It can be calculated by
completing a square.
$$
...
...
@@ -214,5 +214,5 @@ $$
\e
nd{align
*
}
$$
This depends on the variance, which is inverted by the Fourier
T
ransform. So since the power is
This depends on the variance, which is inverted by the Fourier
t
ransform. So since the power is
invariant, the normalization cannot in general be conserved.
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