The document discusses the mathematical relationships between the Fourier and Hartley transforms. It begins by introducing the transforms and their kernel functions. The Fourier kernel is complex while the Hartley kernel is real, making Hartley transforms simpler. It then proves that the Fourier transform equations can be written in terms of the Hartley transform components. Code examples demonstrate the transforms have the same results for scaling and modulation properties. In conclusion, the Fourier and Hartley transforms are mathematically equivalent despite differences in their kernel functions.

2. Mathematics and Applications
of the Twin
" Hartley and Fourier Transforms"
i. Introduction
At the end of the eighteenth century and beginning of the nineteenth century the world around us
have been changed , the new era of scientific renaissance has been birthed with the emergence of
the transform methods. These methods is the basis of many of techniques and discoveries at the
present time because of its capabilities to transfer the functions between different worlds. and one
of the famous methods are Fourier Transform and its successor Hartley Transform which are the
subject of our present .
The Fourier ( by Joseph Fourier) and Hartley (by R. V. L. Hartley) transform methods are integral
methods with a kernel functions that transform the functions (signals) from time domain to
frequency domain , Based on the fact that " any function in our life is just a series of combination
of sinusoidal signals or complex exponential ". as shown in figure (1) below .
Aperiodic function Periodic function
Figure(1) func ons in time and frequency domains
So , the Hartley and Fourier Transform methods in its general concept is just a way that "change our
sight to a function from the front view (along the time domain) to the side view (along the
frequency domain)" . in fact they work like a machine that sucks the whole function (domain , range
and properties ) as an input and manipulates it by its heart or its engine part that called the "kernel
function" and then produces a new function as an output with a new (domain , range and
properties ) , however these functions are related to each other where the changes in each one
means changes in the other .
On the other hand , the complexity of these transformations are depending on the complexity of
the input and kernel functions . this means ,the complexity of transform is associated with the
number of the domains (dimensions) of these function whenever the dimensions (domains) are
increased the complexity will also increases .
In this report we will address to the relations between Fourier and Hartley transform by doing some
comparisons and proofs that lead to consideration these transform methods are a twin and at the
end we will present some of the benefits that be obtained from these methods .

3. ii. FT and HT formulas (Relation and Proofs )
In this section we will address to the Kernel Function of both Fourier and Hartley transform method
and the relation between FT and HT Equation and we will proof the capability of writing the FT
equations (General , Amplitude and Phase equations ) in terms of HT equation components , and in
the rest part of these section we will try to prove (mathematically and by using MATLAB
implementation codes) that the FT and HT outputs and most of their properties (like Scallimg and
Modulation ) are the same .
 Kernel Function
The Hartley transform is purely real and fully equivalent to the well‐known Fourier transform. It is
an offshoot of the Fourier transform with the same physical significance as that of its progenitor.
The two transforms are closely related . So both Fourier and Hartley transforms furnish at each
frequency a pair of numbers that represent a physical oscillation in amplitude and phase and in tum
they give the same information substantiates that the Hartley transform can equally be applicable
to all fields where the Fourier transform is currently being used.
But the main difference between them is the nature of kernel function . as shown below
Fourier Transform Equation Hartley Transform Equation
F w f t e dt
H w f t cas wt dt
Where the kernel function of Fourier Transform (e as shown in figure (2) is a complex function
which needs of four dimension (real , imaginary domains & magnitude , phase ranges) to
represented it and based on the first section of this report , we mentioned that "the complexity of
the transform method is increased when the complexity of kernel function is increased " , hence
the solving by using this method will be more complicated .
Figure (2) Fourier Transform Kernel (e
While the kernel function of Hartley transform (cas wt ) as shown in figure (3) is a real function
that consist of one domain and range and can be represented in plane coordinate , hence the
solving by using this method will be more simplicity than Fourier .
Figure (3) Hartley Transform Kernel (cas wt cos wx sin wx )

4.  FT and HT related equations (proofs)
The well‐known Fourier Transform (FT) general equation can be defined as :
Where R(w) and I(w) are the real and negative imaginary components of the Fourier Transform .
while the Hartley Transform (HT) general equation can be defined as :
Where E(w) and O(w) are the even and odd components of Hartley Transform which can be
expressed by other ways as shown in the two sections (A & B) below :
A‐ Even component
2 ∗
2
B‐ Odd component
2 ∗
2

5. and based on what we mentioned previously (The FT and HT methods close to each other and they
work as a Twin ) , we will find that :
1‐ the ( ) of Fourier Transform equal to the ( ) of Hartley Transform .
2‐ and the (‐ ) equal to the ( ) .
Hence the FT equation can be expressed in terms of the HT components as shown below
2
∗
2
And the same thing for Fourier Amplitude and Phase equations , we can expressed it in terms of
Hartley transform components as shown in section (C & D) below :
C‐ For amplitude
∗ 2 where
∗ 2 where
1
4
∗ 2
1
4
∗ 2
1
4
2 2
1
4
2 2
2
By taking the square root we will get the same result of the equation above:
D‐ For phase

 /
But , Fourier and Hartley phases differ by a constant due to the presence of the negative sign and
hence they can be related as:
4

6.
 FT & HT magnitude result similarity‐Example
In this part , we will try to find the FT and HT of the square
signal f(t) as shown in the figure(4) ,then we will compare
between the results that will show the similarity of their
results .
At the end , we are sketching the results by using MATLAB
as shown in figure (5) and (6) .
1 1 1
0
‐ In Fourier Transform:
1 ∗
1
1
1
1
2
2
2
sin 2 ∗
sin
2 ∗
‐ In Hartley Transform:
cos sin 1 ∗ cos 1 ∗ sin
sin 1
1
cos 1
1
1
sin sin cos cos
1
sin sin cos cos
1
sin sin cos cos
1
2 ∗ sin 2 ∗
sin
2 ∗
Figure(5) Figure(6)
Figure(4)

7.  MATLAB Implementation of (Scaling and Modulation properties )
 Scaling property [h( )=a H(aw)]
Fourier Transform‐Codes Hartley Transform‐Codes
clc; clear all; close all;
%----------- function f(t)-----------
t=-4 : pi/180 : 4;
f = zeros(size(t));
f=heaviside(t+2)-heaviside(t-2);
figure(2);
subplot(2,1,1); plot(t,f,'LineWidth',2);
xlabel('t (sec)');
ylabel('f(t)');
axis([-4 4 0 1.5])
title('Function of t where a=1');
grid
%-------------Fourier transform----------
--
omega = [-50 : 0.1 : 50];
F = zeros(size(omega));
for i = 1 : length(omega)
F(i) = trapz(t,f.*exp(-1i*omega(i)*t));
%Fourier Transform
end
F_magnitude = abs(F); %Magnitude of the
Fourier Transform
subplot(2,1,2);
plot(omega,F_magnitude,'LineWidth',2);
xlabel('omega');
ylabel('|F(jomega)|');
title('Fourier Transform Magnitude');
grid
clear all; clc
%----------- function f(t)-----------
t=-4 : pi/180 : 4;
f = zeros(size(t));
f=heaviside(t+1)-heaviside(t-1);
figure(2);
subplot(2,1,1); plot(t,f,'LineWidth',2);
xlabel('t (sec)');
ylabel('f(t)');
title('Function of t when a=1/2');
axis([-4 4 0 1.5])
grid
%-------------Hartley Transform------------
omega = [-50 : 0.1 : 50];
F = zeros(size(omega));
for i = 1 : length(omega)
F(i) =
trapz(t,f.*(cos(omega(i)*t)+sin(omega(i)*t)))
; %Hartley Transform integral
end
F_magnitude = abs(F); %Magnitude of the
Hartley Transform
subplot(2,1,2);
plot(omega,F_magnitude ,'LineWidth',2);
xlabel('omega ');
ylabel('|F(jomega)|');
title('Hartley Transform Magnitude');
grid
Fourier Transform Hartley Transform

8.
 Modulation property
Carrier and Signal‐Codes Carrier and Signal ‐ sketch
%----------- Carrier ----------%
t = -2*pi : pi/180 : 2*pi;
fs=cos(10*t);
subplot(2,1,1);
plot(t,fs,'LineWidth',2);
xlabel('t (sec)');
ylabel('carrier');
axis([-4 4 -1.5 1.5])
title('Carrier Signal [ cos(10t) ]');
grid
%---------- Signal -----------%
fc = zeros(size(t));
fc=heaviside(t+1)-heaviside(t-1);
subplot(2,1,2);
plot(t,fc,'LineWidth',2);
axis([-4 4 0 1.5])
xlabel('t (sec)');
ylabel('signal');
title('Square signal [ u(t+1)-u(t-
1)]');
grid

9. Fourier Modulation‐Codes Hartley Modulation‐Codes
%----------- Modulation -----------
figure;
t = -2*pi : pi/180 : 2*pi;
f1=heaviside(t+1)-heaviside(t-1);
f=f1.*cos(10*t);
figure(1);
subplot(2,1,1); plot(t,f,'LineWidth',2);
xlabel('t (sec)');
ylabel('f(t)');
title('Function of cos(10t)*u(t+1)-u(t-
1)');
axis([-4.2 4.2 -1.2 1.2 ])
grid
%-------------Fourier transform-----------
omega = [-50 : 0.1 : 50];
F = zeros(size(omega));
for i = 1 : length(omega)
F(i) = trapz(t,f.*exp(-1i*omega(i)*t));
%Fourier Transform
end
F_magnitude = abs(F); %Magnitude of the
Fourier Transform
subplot(2,1,2);
plot(omega,F_magnitude,'LineWidth',2);
xlabel('omega (rad/sec)');
ylabel('|F(jomega)|');
title('Fourier Transform Magnitude');
grid
%----------- Modulation -----------
figure;
t = -2*pi : pi/180 : 2*pi;
f=f1.*cos(10*t);
subplot(2,1,1); plot(t,f,'LineWidth',2);
title(' after Modulation');
xlabel('t (sec)');
ylabel('f(t)');
axis([-4.2 4.2 -1.2 1.2 ])
grid
%-------------Hartley Transform------------
omega = [-50 : 0.1 : 50];
F = zeros(size(omega));
for i = 1 : length(omega)
F(i) =
trapz(t,f.*(cos(omega(i)*t)+sin(omega(i)*t))
); %Hartley Transform integral
end
F_magnitude = abs(F); %Magnitude of the
Hartley Transform
subplot(2,1,2);
plot(omega,F_magnitude ,'LineWidth',2);
xlabel('omega (rad/sec)');
ylabel('|F(jomega)|');
title('Hartley Transform Magnitude');
grid
Fourier Modulation Hartley Modulation

10. iii. Benefits Gained
The Fourier and Hartley Transform Methods are changing our way of thinking and our conception
about interpretation some natural phenomena , also they enabled us to develop and create new
techniques that support our life and activities , in this section we will mention briefly the idea
beyond our hearing and sight senses , and how the engineers benefitted from the FT and HT
methods to create and develop the TV and Phones systems .
A‐ Sense of Hearing
After discovery of FT and HT , we could understand that one of the functions in our ear system is
transformtion the Acoustic Signal from time domain to frequency domain , and these frequencies
are realized by sensor organs are called (Corti) that have a limited capabilities where they can react
with just band of frequencies from 20Hz to 20KHz and this is the limitations of our hearing . as
shown in figure (7).
Figure(7) Sense of Hearing
B‐ Sense of Sight
The same thing for our sight sense where our eyes play the machine role where one of its important
function is transform the signal from the time to frequency domain then realization it by the "Cone
and Rod" organs which can distinguish only a band of frequencies called the band of the visible
waves . the figure (8) shows the inner structure of the eye while the figure (9) shows the spectrum
band of the visible waves .
Figure(8) inner structure of the eye

11.
Figure(10) visible waves
C‐ TV and Phone Systems
Based on the FT and HT transform methods concept and its property that is called (Modulation
Property) , the engineers became able to send many TV channels or Phone Calls as a complex signal,
By modulating the information of it on the carrier signals with different frequencies are separated
from each other by Guard Band . for the TV system the signal is received by LNB which transfer it via
the cable to the receiver that transform it to the frequency domain where each channel has a
certain band of frequency in this spectrum ,hence the person can move from channel to other by
changing from frequency to other . the figure(11) show the idea beyond the TV system . for the
Phone Calls the complex signal will reach to the Switch device inside the call center that transform it
to a group of frequencies then separate each frequency and resend it as a temporal signal to the
des na on . the figure(12) show the Phone Calls idea .
Figure(10) TV System
Figure(11) Phone Calls

12. iv. Conclusion
 Since , the transform methods (FT and HT) have been discovered our life is changing , we
able to understand some of phenomena around us , and we could exploit these method to
create and develop and get a new and modern techniques.
 The essential differences between these two transforms is that Fourier transform gives rise
to complex plane even for real data whereas the Hartley transform always gives the real
plane for real data.
 The properties of Fourier and Hartley transforms are almost the same.
v. References
1. ALEXANDER D. POULARIKAS , "TRANSFORMS and APPLICATIONS HANDBOOK , Third Edition" ,
CRC Press ,Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300 , chapter 4 , p
180‐184 .
2. N. SUNDARARAJAN , "FOURIER AND HARTLEY TRANSFORMS ‐ A MATHEMATICAL TWIN " , Indian
J. pure appl. Math., 28(10) : 1361‐1365, October 1997 , p1‐5 .
3. University of Wisconsin–Madison , Department of Mathematics , Lecture Notes , "
https://www.math.wisc.edu/ ~angenent/Free‐Lecture‐Notes/freecomplexnumbers.pdf " .
4. University of Haifa, The Department of Computer Science, Lecture Notes , "
https://www.math.wisc.edu/ ~angenent/Free‐Lecture‐Notes/freecomplexnumbers.pdf " .
5. Chris Solomon ,Toby Breckon, "Fundamentals of Digital Image Processing",Chichester, West
Sussex, PO19 8SQ, UK , 2011, sec on 1.1 , p 20‐25.
6. Hartley transform , From Wikipedia, the free encyclopedia , "
https://en.wikipedia.org/wiki/Hartley_transform "