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This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.Lecture 6:Continuous Time Fourier Transform (CTFT)
Summary: Extends the notion of the frequency response of a system to the frequency content of a signal. Widely used tool in many areas (communications, control, signal processing, X-ray diffraction, Medical imaging — CAT & PET scan). Continue development of Fourier transform pairs. Illustrate different methods for finding Fourier transforms.
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Lecture #6:
CONTINUOUS TIME FOURIER TRANSFORM (CTFT)
Motivation:
- Extends the notion of the frequency response of a system to the frequency content of a signal.
- Widely used tool in many areas (communications, control, signal processing, X-ray diffraction, Medical imaging — CAT & PET scan).
- Continue development of Fourier transform pairs.
- Illustrate different methods for finding Fourier transforms
Outline:
- The continuous time Fourier transform (CTFT)
- Properties of the CTFT
- Simple CTFT pairs
- Fourier transform pairs
– Fourier transform of the unit step function
– Fourier transform of causal sinusoids
– Fourier transform of rectangular pulse — the sinc function
– Fourier transform of triangular pulse
- Filtering the ECG revisited
- Conclusion
Example — How to filter the ECG?
The recorded activity from the surface of the chest includes the electrical activity of the heart plus extraneous signals or “noise.” How can we design a filter that will reduce the noise?
 Figure 1 |
It is most effective to compute the frequency content of the recorded signal and to identify those components that are due to the electrical activity of the heart and those that are noise. Then the filter can be designed rationally. This is one of many motivations for understanding the Fourier transform.
I. THE CONTINUOUS TIME FOURIER TRANSFORM (CTFT)
1/ Definition
The continuous time Fourier transform of x(t) is defined as
and the inverse transform is defined as
2/ Relation of Fourier and Laplace Transforms
The bilateral Laplace transform is defined by the analysis formula
and the inverse transform is defined by the synthesis formula
Now if the jω axis is in the region of convergence of X(s), then we can substitute s = jω = j2πf into both relations to obtain
3/ Form of the Fourier transform
Finally, by canceling j2π and changing the variable of integration from j2πf to f we obtain
It is clumsy to write X(j2πf). Therefore, from this now on, we rewrite the function X(j2πf) described above as in a simpler form X(f) such that the transform pair would be symmetrical.
X(j2πf) Þ X(f)
The new Fourier transform pair will be in the form
4/ Notation
Notation for the Fourier transform varies appreciably from text to text and in different disciplines. Another common notation is to define the Fourier transform in terms of jω as follows
Differences in notation are largely a matter of taste and different notations result in different locations of factors of 2π. The notation we use minimizes the number of factors of 2π that appear in the expressions we will use in this subject and makes the duality of the Fourier transform with its inverse more transparent.
5/ Why bother with the Fourier transform?
- There are certain simple time functions which are more readily represented by Fourier transforms than by Laplace transforms, e.g., x(t) = 1, x(t) = cos(2πft), periodic time functions, etc.
- Certain important operations on signals are more readily analyzed with Fourier transforms, e.g., sampling, modulation, filtering.
- Examination of both signals and systems in the frequency domain gives insights that complement those obtained in the “time” domain.
6/ Functions that have Laplace transforms but not Fourier transforms
There are some time functions that have a Laplace transform but not a Fourier transform, namely those for which the jω-axis is not inside the region of convergence. For example, x(t) = eαtu(t) for α > 0.
 Figure 2 |
II. PROPERTIES OF THE CTFT
1/ Properties — symmetry
We start with the definition of the Fourier transform of a real time function x(t) and expand both terms in the integrand in terms of odd and even components.
The odd components of the integrand contribute zero to the integral. Hence, we obtain
where
We can infer symmetry properties of the Fourier transform of a real time function x(t).
Therefore, if
x(t) X(f)
Real and even function of t Real and even function of f
Real and odd function of t Imaginary and odd function of f
The angle can be computed as follows,
But, since ±n2π can always be added to the angle, and since is an odd function of f,
Therefore, ÐX(f) is an odd function of f.
2/ Properties — duality
The Fourier transform and its inverse differ only by a sign in the exponent,
Therefore, if x(t)X (f) then X(t)x(−f). This means that if we have found one Fourier transform pair, we automatically know another.
3/ List of simple properties
Some of the important properties are summarized here; a more complete list is appended.
 Figure 3 |
a/ Properties — linearity
Most proofs of Fourier transform properties are simple.
The proof follows from the definition of the Fourier transform as a definite integral.
b/ Delay by to
This result can be seen using the synthesis formula.
Mnemonic: a delay of the time function multiplies the Fourier transform by a lag factor, i.e., a delay of the time function of to adds −2πfto to the angle of the Fourier transform but does not affect the magnitude.
c/ Differentiate in t
This result can be seen using the synthesis formula.
Differentiating the time function, adds π/2 radians to the angle of the Fourier transform and multiplies the magnitude by 2πf.
d/ Multiply by ej2πfot
This result can be seen using the analysis formula.
This result can also be obtained from the delay-in-time property and duality. Mnemonic: multiplying a time function by a complex exponential at frequency fo shifts the Fourier transform to fo.
e/ Convolution in time
If x(t) = x1(t) ∗ x2(t) then X(f) is obtained as follows