Assume that we have a generalized, timelimited pulse centered at t = 0 as shown below.
The Fourier Transform of this pulse can be developed by starting with a periodic version of this pulse where the original pulse now repeats every T seconds.
Note: _{}
f_{T}(t) is periodic with period T so we can express it by its exponential Fourier series as
_{}
where
_{}
and
_{}
Now let’s make a small change in notation
1. w_{n} = n*w_{0}
2. F(w_{n}) = T*F_{n}
We now have
_{} and _{}
The sum can be rewritten as
_{}
or
_{}
Taking the limit as T ¥
_{}
But w_{0} = 2p/T so for large T let w_{0} Dw and the limit becomes
_{}
or since T ¥ implies that Dw 0 and the sum, in the limit, becomes an integral
_{}and _{}
This pair of equations defines the Fourier Transform
1. F(w) is the Fourier Transform of f(t)
2. f(t) is the inverse Fourier Transform of F(w)
3. F(w) is also called the Spectral Density of f(t) as it describes how the energy of the original pulse is distributed as a function of frequency (in radians per second)
I use a backwards upper case script “F” to denote taking the Fourier Transform of a function and the same symbol with a “1” superscript to denote taking the inverse Fourier Transform.
Take the Fourier Transform of the singlesided exponential
_{}
_{}
_{}
_{}
_{}
Note that the Fourier Transform is complex. It has a magnitude and a phase. The magnitude is found by multiplying it by its complex conjugate and taking the square root.
_{}
_{}
_{} This is the magnitude
Now find the phase. First, find the real and imaginary parts.
_{}
_{}
_{}
Therefore the real part is
_{}
and the imaginary part is
_{}
The phase is then given by
_{}
We run into special functions
when taking the Fourier Transform of functions that have infinite energy. The first of these special functions is the Delta Function
_{}
Where G_{e}(t) is any function from the set of all functions having the properties
1. _{}
2. _{} For all t ¹ 0
Integrating the product of the Delta Function with a “wellbehaved” function results in “sampling” the “wellbehaved” function at the time that the Delta Function goes to infinity. Or
_{}
Proof
Use Integration by parts
_{}
Let U(t) = f(t) and dV(t) = d(tt_{0})dt
_{}
Case 1: a < t_{0} < b
_{}
_{}
_{}
_{} Q.E.D
Case 2: t_{0} < a or t_{0} > b
_{} Q.E.D
Take the Fourier Transform of a constant
_{}
Here the integral can’t be directly computed, we have to approach it as a limiting case. Let’s replace the constant with a parameterized function that equals the constant as its parameter approaches zero, the doublesided exponential function:
_{}
Now the Transform becomes:
_{}
Let u = w in the first integral
_{}
From our first example this is:
_{}
Now we need to take the limit as a 0 to get F(w)
_{}
_{}
so this is a dfunction that goes to ¥ at w = 0 if its integral is a constant.
_{}
Let a*x = w
_{}
_{}
_{}
_{}
_{}
Therefore
_{}
1: Find the Fourier Transforms for each of the two pulses
2: Find the transfer function for the simple RC lowpass filter

3: Determine the Fourier Transform of the RC lowpass filter output due to each of the pulses in part 1
4: Find the limit of each of the results in part 3 as Dt 0
If f(t) F(w)
Then F(t) 2p f(w)
Proof:
_{}
Therefore
_{}
Let u = w and v = t
_{}
Now let w = v and t = u
_{}
Therefore F(t) 2p f(w)
And if f(t) is an even function
F(t) 2p f(w)
If f_{1}(t) F_{1}(w)
And f_{2}(t) F_{2}(w)
Then [a*f_{1}(t) + b*f_{1}(t)] [a*F_{1}(w) + b*F_{2}(w)]
Proof:
Results due to the linearity of integration
If f(t) F(w)
Then for a real
f(a*t) _{}
Proof:
_{}
case 1: a > 0 Let x = a*t
_{}
_{}
or
_{}
case 2: a < 0 Again let x = a*t
_{} (Note the limits are now backwards)
_{}
or
_{}
Therefore including both cases
f(a*t) _{}
Q. E. D.
Note: The compression of a function in the time domain results in an expansion in the frequency domain and vice versa.
If _{}
Then _{}
Proof:
_{}_{}
_{}
_{}
or
_{}
Q. E. D.
Note: The Modulation Theorem (very important in communications)
Remember Euler’s Identities
_{} and _{}
therefore
_{}
or
_{}
similarly
_{}
or
_{}
If _{}
Then _{}
Proof:
_{}
_{}
_{}
_{}
Q. E. D._{}
If _{}
Then _{}
And _{}
Proof:
First for differentiation (part 1)
_{}
_{}
_{}
_{}
_{}
or
_{} Q.
E. D. for part 1
Now for integration (part 2)
_{}
_{}
Interchanging the order of integration
_{}
_{}
_{}
or
_{} Q. E. D. for part 2
If _{}
Then _{}
Proof:
_{}
_{}
_{}
_{}
or
_{} Q. E. D.
Definition: the convolution of two functions _{} is defined as:
_{}
Time Convolution
If _{}
And _{}
Then _{}
Proof:
Therefore
_{}
_{}
Let u = t  t in the inner integral
_{}
_{}
Since the inner integral is no longer a function of t, it can be brought out as a constant and this leaves
_{}
or
_{} Q. E. D
Frequency Convolution
If _{}
And _{}
Then _{}
Proof: Same method
as for time convolution