Chapter 8. Continuous Time Fourier Transform (CTFT)
8.1. Continuous Time Aperiodic Signals*
Introduction
This module describes the type of signals acted on by the Continuous Time Fourier Transform.
Relevant Spaces
The Continuous-Time Fourier Transform maps infinite-length (a-periodic), continuous-time signals in L 2 to infinite-length, discrete-frequency signals in l 2 .
Periodic and Aperiodic Signals
When a function repeats itself exactly after some given period, or cycle, we say it's periodic. A periodic function can be mathematically defined as: f(t) = f(t + mT) m ∈ ℤ where T > 0 represents the fundamental period of the signal, which is the smallest positive value of T for the signal to repeat. Because of this, you may also see a signal referred to as a T-periodic signal. Any function that satisfies this equation is said to be periodic with period T.
An aperiodic CT function f(t) does not repeat for any T ∈ ℝ ; i.e. there exists no T such that this equation holds.
Suppose we have such an aperiodic function f(t) . We can construct a periodic extension of f(t) called f To(t) , where f(t) is repeated every T 0 seconds. If we take the limit as T 0 → ∞ , we obtain a precise model of an aperiodic signal for which all rules that govern periodic signals can be applied, including Fourier Analysis (with an important modification). For more detail on this distinction, see the module on the Continuous Time Fourier Transform.
Aperiodic Signal Demonstration
Figure 8.1.
Conclusion
Any aperiodic signal can be defined by an infinite sum of periodic functions, a useful definition that makes it possible to use Fourier Analysis on it by assuming all frequencies are present in the signal.
8.2. Continuous Time Fourier Transform (CTFT)*
Introduction
In this module, we will derive an expansion for any arbitrary continuous-time function, and in doing so, derive the Continuous Time Fourier Transform (CTFT).
Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system ℋ given ⅇ st as an input amounts to simple multiplication, where H(s) ∈ ℂ is the eigenvalue corresponding to s. As shown in the figure, a simple exponential input would yield the output y(t) = H(s)ⅇ st
Using this and the fact that ℋ is linear, calculating y(t) for combinations of complex exponentials is also straightforward.
c
1
ⅇ
s
1
t
+ c
2
ⅇ
s
2
t
→ c
1
H(s
1)ⅇ
s
1
t
+ c
2
H(s
2)ⅇ
s
2
t
The action of H on an input such as those in the two equations above is easy to explain. ℋ independently scales each exponential component ⅇ s n t by a different complex number H(s n ) ∈ ℂ . As such, if we can write a function f(t) as a combination of complex exponentials it allows us to easily calculate the output of a system.
Now, we will look to use the power of complex exponentials to see how we may represent arbitrary signals in terms of a set of simpler functions by superposition of a number of complex exponentials. Below we will present the Continuous-Time Fourier Transform (CTFT), commonly referred to as just the Fourier Transform (FT). Because the CTFT deals with nonperiodic signals, we must find a way to include all real frequencies in the general equations. For the CTFT we simply utilize integration over real numbers rather than summation over integers in order to express the aperiodic signals.
Fourier Transform Synthesis
Joseph
Fourier demonstrated that an arbitrary
s(t)
can be written as a linear combination of harmonic
complex sinusoids
where
is the fundamental frequency. For almost all
s(t)
of practical interest, there exists
c
n
to make Equation true. If
s(t)
is finite energy (
s(t) ∈ L
2[0, T]
), then the equality in Equation
holds in the sense of energy convergence; if
s(t)
is continuous, then Equation holds
pointwise. Also, if
s(t)
meets some mild conditions (the Dirichlet
conditions), then Equation holds
pointwise everywhere except at points of discontinuity.
The c n - called the Fourier coefficients - tell us "how much" of the sinusoid ⅇ j ω 0 n t is in s(t) . The formula shows s(t) as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set of complex exponentials {ⅇ j ω 0 n t , n ∈ ℤ } form a basis for the space of T-periodic continuous time functions.
Equations
Now, in order to take this useful tool and apply it to arbitrary non-periodic signals, we will have to delve deeper into the use of the superposition principle. Let
s
T
(t)
be a periodic signal having period
T
.
We want to consider what happens to this signal's spectrum as the period goes to infinity. We denote the spectrum for any assumed value of the period by
c
n
(T).
We calculate the spectrum according to the Fourier formula for a periodic signal, known as the Fourier Series (for more on this derivation, see the section on Fourier Series.) 
where 
and where we have used a symmetric placement of the integration interval about the origin for subsequent derivational convenience. We vary the frequency index
n
proportionally as we increase the period. Define
making the corresponding Fourier Series
As the period increases, the spectral lines become closer together, becoming a continuum. Therefore,
with
Warning
It is not uncommon to see the above formula written slightly different. One of the most common differences is the way that the exponential is written. The above equations use the radial frequency variable Ω in the exponential, where Ω = 2π f , but it is also common to include the more explicit expression, ⅈ2πft , in the exponential. Click here for an overview of the notation used in Connexion's DSP modules.
Example 8.1.
We know from Euler's formula that 
CTFT Definition Demonstration
Figure 8.2.
Example Problems
Exercise 8.2.2. (Go to Solution)
Find the inverse Fourier transform of the ideal lowpass filter
defined by
Fourier Transform Summary
Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.
The continuous time Fourier series analysis formula gives the coefficients of the Fourier series expansion.
In both of these equations
is the fundamental frequency.
Solutions to Exercises
Solution to Exercise 8.2.1. (Return to Exercise)
In order to calculate the Fourier transform, all we need
to use is Equation 8.1, complex exponentials,
and basic calculus.
Solution to Exercise 8.2.2. (Return to Exercise)
Here we will use Equation 8.2 to
find the inverse FT given that
t ≠ 0
.
8.3. Common Fourier Transforms*
Common CTFT Properties
| Time Domain Signal | Frequency Domain Signal | Condition |
|---|---|---|
| ⅇ – (at) u(t) |
| a > 0 |
| ⅇ at u( – t) |
| a > 0 |
| ⅇ – (a|t|) |
| a > 0 |
| t ⅇ – (at) u(t) |
| a > 0 |
| t n ⅇ – (at) u(t) |
| a > 0 |
| δ(t) | 1 | |
| 1 | 2π δ(ω) | |
| ⅇ ⅈ ω 0 t | 2π δ(ω − ω 0) | |
| cos(ω 0 t) | π(δ(ω − ω 0) + δ(ω + ω 0)) | |
| sin(ω 0 t) | ⅈ π(δ(ω + ω 0) − δ(ω − ω 0)) | |
| u(t) |
| |
| sgn(t) |
| |
| cos(ω 0 t)u(t) |
| |
| sin(ω 0 t)u(t) |
| |
| ⅇ – (at)sin(ω 0 t)u(t) |
| a > 0 |
| ⅇ – (at)cos(ω 0 t)u(t) |
| a > 0 |
| u(t + τ) − u(t − τ) |
| |
| u(ω + ω 0) − u(ω − ω 0) | |
|
| |
|
| |
|
|
|
|
|
triag[n] is the triangle function for arbitrary real-valued n .
8.4. Properties of the CTFT*
Introduction
This module will look at some of the basic properties of the Continuous-Time Fourier Transform (CTFT).
Note
We will be discussing these properties for aperiodic, continuous-time signals but understand that very similar properties hold for discrete-time signals and periodic signals as well.
Discussion of Fourier Transform Properties
Linearity
The combined addition and scalar multiplication properties in the table above demonstrate the basic property of linearity. What you should see is that if one takes the Fourier transform of a linear combination of signals then it will be the same as the linear combination of the Fourier transforms of each of the individual signals. This is crucial when using a table of transforms to find the transform of a more complicated signal.
Example 8.2.
We will begin with the following signal: z(t) = a f 1(t) + b f 2(t) Now, after we take the Fourier transform, shown in the equation below, notice that the linear combination of the terms is unaffected by the transform. Z(ω) = a F 1(ω) + b F 2(ω)
Symmetry
Symmetry is a property that can make life quite easy when solving problems involving Fourier transforms. Basically what this property says is that since a rectangular function in time is a sinc function in frequency, then a sinc function in time will be a rectangular function in frequency. This is a direct result of the similarity between the forward CTFT and the inverse CTFT. The only difference is the scaling by 2π and a frequency reversal.
Time Scaling
This property deals with the effect on the frequency-domain representation of a signal if the time variable is altered. The most important concept to understand for the time scaling property is that signals that are narrow in time will be broad in frequency and vice versa. The simplest example of this is a delta function, a unit pulse with a very small duration, in time that becomes an infinite-length constant function in frequency.
The table above shows this idea for the general transformation from the time-domain to the frequency-domain of a signal. You should be able to easily notice that these equations show the relationship mentioned previously: if the time variable is increased then the frequency range will be decreased.
Time Shifting
Time shifting shows that a shift in time is equivalent to a linear phase shift in frequency. Since the frequency content depends only on the shape of a signal, which is unchanged in a time shift, then only the phase spectrum will be altered. This property is proven below:
Example 8.3.
We will begin by letting
z(t) = f(t − τ)
.
Now let us take the Fourier transform with the previous
expression substituted in for
z(t)
.
Now let us make a simple change of variables, where
σ = t − τ
. Through the calculations below, you can see
that only the variable in the exponential are altered thus
only changing the phase in the frequency domain.
Convolution
Convolution is one of the big reasons for converting signals to the frequency domain, since convolution in time becomes multiplication in frequency. This property is also another excellent example of symmetry between time and frequency. It also shows that there may be little to gain by changing to the frequency domain when multiplication in time is involved.
We will introduce the convolution integral here, but if you have not seen this before or need to refresh your memory, then look at the continuous-time convolution module for a more in depth explanation and derivation.
Time Differentiation
Since LTI systems can be represented in terms of differential equations, it is apparent with this property that converting to the frequency domain may allow us to convert these complicated differential equations to simpler equations involving multiplication and addition. This is often looked at in more detail during the study of the Laplace Transform.
Parseval's Relation
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.
Figure 8.3.
Modulation (Frequency Shift)
Modulation is absolutely imperative to communications applications. Being able to shift a signal to a different frequency, allows us to take advantage of different parts of the electromagnetic spectrum is what allows us to transmit television, radio and other applications through the same space without significant interference.
The proof of the frequency shift property is very similar to
that of the time
shift; however, here we would use the inverse Fourier
transform in place of the Fourier transform. Since we went
through the steps in the previous, time-shift proof, below
we will just show the initial and final step to this proof:
Now we would simply reduce this equation through another
change of variables and simplify the terms. Then we will
prove the property expressed in the table above:
z(t) = f(t)ⅇ
ⅈφt
Properties Demonstration
An interactive example demonstration of the properties is included below:
Figure 8.4.
Summary Table of CTFT Properties
| Operation Name | Signal ( f(t) ) | Transform ( F(ω) ) |
|---|---|---|
| Linearity | a(f 1, t) + b(f 2, t) | a(F 1, ω) + b(F 2, ω) |
| Scalar Multiplication | α f(t) | α F(ω) |
| Symmetry | F(t) | 2π f( – ω) |
| Time Scaling | f(αt) |
|
| Time Shift | f(t − τ) | F(ω)ⅇ – (ⅈωτ) |
| Convolution in Time | (f 1(t), f 2(t)) | F 1(t)F 2(t) |
| Convolution in Frequency | f 1(t)f 2(t) |
|
| Differentiation |
| (ⅈω) n F(ω) |
| Parseval's Theorem |
|
|
| Modulation (Frequency Shift) | f(t)ⅇ ⅈφt | F(ω − φ) |
8.5. Continuous Time Convolution and the CTFT*
Introduction
This module discusses convolution of continuous signals in the time and frequency domains.
Continuous Time Fourier Transform
The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.
Convolution Integral
The convolution integral expresses the output of an LTI system based
on an input signal,
x(t)
, and the system's impulse response,
h(t)
. The
convolution integral is expressed as
Convolution is such an important tool that it is represented
by the symbol
*
, and can be written as
y(t) = x(t) * h(t)
Convolution is commutative. For more information on the characteristics of the convolution
integral, read about the Properties of Convolution.
Demonstration
Figure 8.5.
Convolution Theorem
Let f and g be two functions with convolution f * g .. Let F be the Fourier transform operator. Then
By applying the inverse Fourier transform F – 1 , we can write:
Conclusion
The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).