Worksheet 7

Worksheet 7#

To accompany Chapter 4.2 Fourier transforms of commonly occurring signals#

This worksheet can be downloaded as a PDF file. We will step through this worksheet in class.

An annotatable copy of the notes for this presentation will be distributed before the second class meeting as Worksheet 7 in the Week 5: Classroom Activities section of the Canvas site. I will also distribute a copy to your personal Worksheets section of the OneNote Class Notebook so that you can add your own notes using OneNote.

You are expected to have at least watched the video presentation of Unit 4.2: Fourier transforms of commonly occurring signals of the notes before coming to class. If you haven’t watch it afterwards!

After class, the lecture recording and the annotated version of the worksheets will be made available through Canvas.

Reminder of the Definitions#

Last time we derived the Fourier Transform by evaluating what would happen when a periodic signal was made periodic. Let us restate the definitions.

The Fourier Transform#

Used to convert a function of time $f (t)$ to a function of radian frequency $F (ω)$ :

F {f (t)} = \int_{- \infty}^{\infty} f (t) e^{- j ω t} d t = F (ω) .

The Inverse Fourier Transform#

Used to convert a function of frequency $F (ω)$ to a function of time $f (t)$ :

F^{- 1} {F (ω)} = \frac{1}{2 π} \int_{- \infty}^{\infty} F (ω) e^{j ω t} d ω = f (t) .

Note, the factor $2 π$ is introduced because we are changing units from radians/second to seconds.

Duality of the transform#

Note the similarity of the Fourier and its Inverse.

This has important consequences in filter design and later when we consider sampled data systems.

Table of Common Fourier Transform Pairs#

This table is adapted from Table 8.9 of Karris. See also: Wikibooks: Engineering Tables/Fourier Transform Table and Fourier Transform—WolframMathworld for more complete references.

	Name	$f (t)$	$F (ω)$	Remarks
1.	Dirac delta	$δ (t)$	$1$	Constant energy at all frequencies.
2.	Time sample	$δ (t - t_{0})$	$e^{- j ω t_{0}}$
3.	Phase shift	$e^{j ω_{0} t}$	$2 π δ (ω - ω_{0})$
4.	Signum	$sgn t$	$\frac{2}{j ω}$	also known as sign function
5.	Unit step	$u_{0} (t)$	$\frac{1}{j ω} + π δ (ω)$
6.	Cosine	$\cos ω_{0} t$	$π [δ (ω - ω_{0}) + δ (ω + ω_{0})]$
7.	Sine	$\sin ω_{0} t$	$- j π [δ (ω - ω_{0}) - δ (ω + ω_{0})]$
8.	Single pole	$e^{- a t} u_{0} (t)$	$\frac{1}{j ω + a}$	$a > 0$
9.	Double pole	$t e^{- a t} u_{0} (t)$	$\frac{1}{(j ω + a)^{2}}$	$a > 0$
10.	Complex pole (cosine component)	$e^{- a t} \cos ω_{0} t u_{0} (t)$	$\frac{j ω + a}{(j ω + a)^{2} + ω_{0}^{2}}$	$a > 0$
11.	Complex pole (sine component)	$e^{- a t} \sin ω_{0} t u_{0} (t)$	$\frac{ω_{0}}{(j ω + a)^{2} + ω_{0}^{2}}$	$a > 0$

Some Selected Fourier Transforms#

The Dirac Delta#

δ (t) \Leftrightarrow 1

Proof: uses sampling and sifting properties of $δ (t)$ .

See OneNote

Fourier transform of the Dirac delta

MATLAB:

format compact; 
setappdata(0, "MKernel_plot_format", 'svg')

syms t omega omega_0 t0;
u0(t) = heaviside(t); % useful utility function

fourier(dirac(t))

ans =
1

δ (t - t_{0}) \Leftrightarrow e^{- j ω t_{0}}

fourier(dirac(t - t0),omega)

ans =
exp(-omega*t0*1i)

DC#

1 \Leftrightarrow 2 π δ (ω)

Fourier transform of a constant (model of DC)

MATLAB:

A = sym(1); % take one to be a symbol
fourier(A,omega)

ans =
2*pi*dirac(omega)

Related by frequency shifting property:

e^{j ω_{0} t} \Leftrightarrow 2 π δ (ω - ω_{0})

Cosine (Sinewave with even symmetry)#

\cos (t) = \frac{1}{2} (e^{j ω_{0} t} + e^{- j ω_{0} t}) \Leftrightarrow π δ (ω - ω_{0}) + π δ (ω + ω_{0})

Fourier transform of a cosine signal

Note: $f (t)$ is real and even. $F (ω)$ is also real and even.

MATLAB:

fourier(cos(omega_0*t),omega)

ans =
pi*(dirac(omega - omega_0) + dirac(omega + omega_0))

Sinewave#

\sin (t) = \frac{1}{j 2} (e^{j ω_{0} t} - e^{- j ω_{0} t}) \Leftrightarrow - j π δ (ω - ω_{0}) + j π δ (ω + ω_{0})

Fourier transform of a sinewave signal

Note: $f (t)$ is real and odd. $F (ω)$ is imaginary and odd.

MATLAB:

fourier(sin(omega_0*t),omega)

ans =
-pi*(dirac(omega - omega_0) - dirac(omega + omega_0))*1i

Signum (Sign)#

The signum function is a function whose value is equal to

\begin{array}{r} sgn t = {\begin{array}{c} - 1 t < 0 \\ 0 x = 0 \\ + 1 t > 0 \end{array} \end{array}

MATLAB:

fourier(sign(t),omega)

ans =
-2i/omega

The transform is:

sgn t = u_{0} (t) - u_{0} (- t) = \frac{2}{j ω}

Fourier transform of sgn(t)

This function is often used to model a voltage comparitor in circuits.

Example 4: Unit Step#

Use the signum function to show that

F {u_{0} (t))} = π δ (ω) + \frac{1}{j ω}

Clue#

Define

sgn t = 2 u_{0} (t) - 1

sgn(t) = 2 u_0(t) - 1

Does that help?

MATLAB:

fourier(u0(t),omega)

ans =
pi*dirac(omega) - 1i/omega

Example 5#

Use the results derived so far to show that

e^{j ω_{0} t} u_{0} (t) \Leftrightarrow π δ (ω - ω_{0}) + \frac{1}{j (ω - ω_{0})}

Hint: linearity plus frequency shift property.

Example 6#

Use the results derived so far to show that

\sin ω_{0} t u_{0} (t) \Leftrightarrow \frac{π}{j 2} [δ (ω - ω_{0}) - δ (ω + ω_{0})] + \frac{ω_{0}}{ω_{0}^{2} - ω^{2}}

Hint: Euler’s formula plus solution to example 5.

Important note: the equivalent example in Karris (Section 8.4.9 Eq. 8.75 pp 8-23—8-24) is wrong!

See worked solution in OneNote for corrected proof.

Example 7#

Use the result of Example 3 to determine the Fourier transform of $\cos ω_{0} t u_{0} (t)$ .

Answer#

\cos ω_{0} t u_{0} (t) \Leftrightarrow \frac{π}{2} [δ (ω - ω_{0}) + δ (ω + ω_{0})] + \frac{j ω}{ω_{0}^{2} - ω^{2}}

Derivation of the Fourier Transform from the Laplace Transform#

If a signal is a function of time $f (t)$ which is zero for $t \leq 0$ , we can obtain the Fourier transform from the Laplace transform by substituting $s$ by $j ω$ .

Example 8: Single Pole Filter#

Given that

L {e^{- a t} u_{0} (t)} = \frac{1}{s + a}

Compute

F {e^{- a t} u_{0} (t)}

Example 9: Complex Pole Pair cos term#

Given that

L {e^{- a t} \cos ω_{0} t u_{0} (t)} = \frac{s + a}{(s + a)^{2} + ω_{0}^{2}}

Compute

F {e^{- a t} \cos ω_{0} t u_{0} (t)}

Fourier Transforms of Common Signals#

We shall conclude this session by computing as many of the the Fourier transform of some common signals as we have time for.

rectangular pulse
triangular pulse
periodic time function
unit impulse train (model of regular sampling)

I have created some worked examples to help with revision: see Canvas Worked Solutions to Selected Week 5 Problems and the OneNote class notebook Fourier Transforms of Common Signals).

Worksheet 7

Contents

Worksheet 7#

To accompany Chapter 4.2 Fourier transforms of commonly occurring signals#

Reminder of the Definitions#

The Fourier Transform#

The Inverse Fourier Transform#

Duality of the transform#

Table of Common Fourier Transform Pairs#

Some Selected Fourier Transforms#

The Dirac Delta#

DC#

Cosine (Sinewave with even symmetry)#

Sinewave#

Signum (Sign)#

Example 4: Unit Step#

Clue#

Example 5#

Example 6#

Example 7#

Answer#

Derivation of the Fourier Transform from the Laplace Transform#

Example 8: Single Pole Filter#

Example 9: Complex Pole Pair cos term#

Fourier Transforms of Common Signals#