1. Math for CS Fourier Transforms
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

2. Background
While the Fourier series/transform is very important for representing a signal in the frequency domain, it is also important for calculating a system’s response (convolution).
A system’s transfer function is the Fourier transform of its impulse response
Fourier transform of a signal’s derivative is multiplication in the frequency domain: jwX(jw)
Convolution in the time domain is given by multiplication in the frequency domain (similar idea to log transformations)

3. Fourier Series in exponential form
Math for CS
Fourier Series in exponential form
Consider the Fourier series of the 2T periodic function:
Due to the Euler formula
It can be rewritten as
With the decomposition coefficients calculated as:
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
(1)
(2)

4. Fourier transform (3) (4) The frequencies are and
Math for CS
Fourier transform
The frequencies are and
Therefore (1) and (2) are represented as
Since, on one hand the function with period T has also the periods kT for any integer k, and on the other hand any non-periodic function can be considered as a function with infinite period, we can run the T to infinity, and obtain the Riemann sum with ∆w→∞, converging to the integral:
(3)
Outline:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
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(4)

5. Fourier transform definition
Math for CS
Fourier transform definition
The integral (4) suggests the formal definition:
The funciotn F(w) is called a Fourier Transform of function f(x) if:
The function
Is called an inverse Fourier transform of F(w).
(5)
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
(6)

6. Example 1 The Fourier transform of is The inverse Fourier transform is
Math for CS
Example 1
The Fourier transform of is
The inverse Fourier transform is
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

7. Math for CS
Fourier Integral
If f(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is absolutely integrable on R, i.e.
converges, then
Remark: the above conditions are sufficient, but not necessary.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

8. Properties of Fourier transform
Math for CS
Properties of Fourier transform
1 Linearity:
For any constants a, b the following equality holds:
Proof is by substitution into (5).
Scaling:
For any constant c, the following equality holds:
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
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Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

9. Properties of Fourier transform 2
Math for CS
Properties of Fourier transform 2
Time shifting:
Proof:
Frequency shifting:
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Central Scientific Problem – Artificial Intelligence
Machine Learning:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

10. Properties of Fourier transform 3
Math for CS
Properties of Fourier transform 3
Symmetry:
Proof:
The inverse Fourier transform is
therefore
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

11. Properties of Fourier transform 4
Math for CS
Properties of Fourier transform 4
Modulation:
Proof:
Using Euler formula, properties 1 (linearity) and 4 (frequency shifting):
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
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Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

12. Differentiation in time
Math for CS
Differentiation in time
Transform of derivatives
Suppose that f(n) is piecewise continuous, and absolutely integrable on R. Then
In particular
and
Proof:
From the definition of F{f(n)(t)} via integrating by parts.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

13. Math for CS
Example 2
The property of Fourier transform of derivatives can be used for solution of differential equations:
Setting F{y(t)}=Y(w), we have
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
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14. Example 2 Then Therefore Math for CS Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

15. Frequency Differentiation
Math for CS
Frequency Differentiation
In particular and
Which can be proved from the definition of F{f(t)}.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
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Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

16. Fourier Transform analysis synthesis
A CT signal x(t) and its frequency domain, Fourier transform signal, X(jw), are related by
This is denoted by:
For example:
Often you have tables for common Fourier transforms
The Fourier transform, X(jw), represents the frequency content of x(t).
It exists either when x(t)->0 as |t|->∞ or when x(t) is periodic (it generalizes the Fourier series)
analysis
synthesis

17. Linearity of the Fourier Transform
The Fourier transform is a linear function of x(t)
This follows directly from the definition of the Fourier transform (as the integral operator is linear) & it easily extends to an arbitrary number of signals
Like impulses/convolution, if we know the Fourier transform of simple signals, we can calculate the Fourier transform of more complex signals which are a linear combination of the simple signals

18. Fourier Transform of a Time Shifted Signal
We’ll show that a Fourier transform of a signal which has a simple time shift is:
i.e. the original Fourier transform but shifted in phase by –wt0
Proof
Consider the Fourier transform synthesis equation:
but this is the synthesis equation for the Fourier transform
e-jw0tX(jw)

19. Example: Linearity & Time Shift
Consider the signal (linear sum of two time shifted rectangular pulses)
where x1(t) is of width 1, x2(t) is of width 3, centred on zero (see figures)
Using the FT of a rectangular pulse L10S7
Then using the linearity and time shift Fourier transform properties t
x1(t)
x2(t)
x (t)

20. Fourier Transform of a Derivative
By differentiating both sides of the Fourier transform synthesis equation with respect to t:
Therefore noting that this is the synthesis equation for the Fourier transform jwX(jw)
This is very important, because it replaces differentiation in the time domain with multiplication (by jw) in the frequency domain.
We can solve ODEs in the frequency domain using algebraic operations (see next slides)

21. Convolution in the Frequency Domain
We can easily solve ODEs in the frequency domain:
Therefore, to apply convolution in the frequency domain, we just have to multiply the two Fourier Transforms.
To solve for the differential/convolution equation using Fourier transforms:
Calculate Fourier transforms of x(t) and h(t): X(jw) by H(jw)
Multiply H(jw) by X(jw) to obtain Y(jw)
Calculate the inverse Fourier transform of Y(jw)
H(jw) is the LTI system’s transfer function which is the Fourier transform of the impulse response, h(t). Very important in the remainder of the course (using Laplace transforms)
This result is proven in the appendix

22. Example 1: Solving a First Order ODE
Calculate the response of a CT LTI system with impulse response:
to the input signal:
Taking Fourier transforms of both signals:
gives the overall frequency response:
to convert this to the time domain, express as partial fractions:
Therefore, the CT system response is:
assume
ba

23. Example 2: Design a Low Pass Filter
Consider an ideal low pass filter in frequency domain:
The filter’s impulse response is the inverse Fourier transform
which is an ideal low pass CT filter. However it is non-causal, so this cannot be manufactured exactly & the time-domain oscillations may be undesirable
We need to approximate this filter with a causal system such as 1st order LTI system impulse response {h(t), H(jw)}: w
H(jw) wc
-wc
h(t) t

24. Math for CS
Convolution
The convolution of two functions f(t) and g(t) is defined as:
Theorem:
Proof:
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.

25. Appendix: Proof of Convolution Property
Taking Fourier transforms gives:
Interchanging the order of integration, we have
By the time shift property, the bracketed term is e-jwtH(jw), so

26. Summary
The Fourier transform is widely used for designing filters. You can design systems with reject high frequency noise and just retain the low frequency components. This is natural to describe in the frequency domain.
Important properties of the Fourier transform are:
1. Linearity and time shifts
2. Differentiation
3. Convolution
Some operations are simplified in the frequency domain, but there are a number of signals for which the Fourier transform does not exist – this leads naturally onto Laplace transforms. Similar properties hold for Laplace transforms & the Laplace transform is widely used in engineering analysis.

27. Lecture 11: Exercises Theory
Using linearity & time shift calculate the Fourier transform of
Use the FT derivative relationship (S7) and the Fourier series/transform expression for sin(w0t) (L10-S3) to evaluate the FT of cos(w0t).
Calculate the FTs of the systems’ impulse responses
a) b)
Calculate the system responses in Q3 when the following input signal is applied
Matlab/Simulink
Verify the answer to Q1 using the Fourier transform toolbox in Matlab
Verify Q3 and Q4 in Simulink
Simulate a first order system in Simulink and input a series of sinusoidal signals with different frequencies. How does the response depend on the input frequency (S12)?