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\title{37335 - Differential Equations Assignment 2}
\author{Zachary Zerafa - 24557656}

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\section{Introduction}


Mathematics is known for its

We will work in a pure mathematical envrionment with the introduction of necessary physical axioms.

The aim of this thesis is to:
- Review concepts in radio technology (necessary theory of physics and electrical engineering)
- Synthesize these concepts into pure mathematical formulations
- Demonstrate the use of this formal system to problems in radio technology


synthesize the current state of radio engineering into pure mathematics. This will require encoding fundamental concepts of physics and electrical engineering axiomatically into so that the pure mathematician can understand radio engineering as a formal system.

The development of radio engineering has relied extensively on the empirical sciences.

There are numerous reasons why one would desire to translate ideas from radio engineering into a pure mathematics envrionment.
A particular focus of this is the recurrence of Fourier analysis in such mathematical formalizations. Fourier analysis plays a crucial role due its dual role in interpreting functions in a frequency space, as well as its ability to solve frequently occuring differential equations; we will use repeatedly use Fourier analysis as a means to both ends.


- Review of nececcary mathematics
- Mathematical representation of radio waves
- Mathematical representation of signal processing


We will now review specific mathematics that will be frequently employed in our analysis.

\section{Fourier analysis}
\section{Hilbert transform}

\[\mathcal{H}\{f\} = \frac{1}{\pi} \int^{\infty}_{-\infty} \frac{f(u)}{x-u} du \]

\[ \mathcal{H}\{\delta\}= \frac{1}{\pi t} \]
\section{Differential equations}
\section{Control theory}

As luck has it, there is an interdicsiplinary field called 'control theory' that will prove extremely effective in 


The idea is to model specific circuits used in the construction of radios will be . Instead of interpreting concrete blueprints for circuits, we can consider black boxes that emulate their ideal mathematical response for any input. Control theory allows a formal system where one can imagine, and theorize revolutions for the field of radio engineering by means of pure mathematics! In practice one would be forced to engineer circuits to replicate (or approximate) the behaviour of dynamical systems, however control theory provides a means to translate physical systems into mathematical systems.

Yet another benefit of a mathematical analysis is that since we consider 'mathematical black boxes', we can consider idealized versions of physical systems to facilitate our although study.


\section{Mathematics of electromagnetic radiation}


Antennae are engineered devices to transmit and receive data by means of electromagnetic radiation. This is produced by an alternating current through an inductor (the antenna), and electromagnetic radiation interacts with conductors, hence how antennae receive signals.

\subsection{Maxwell's equations}

\[ \nabla \times \mathbf{E} = -\frac{d\mathbf{B}}{t} \]
\[ \nabla \times \mathbf{H} = \frac{d\mathbf{D}}{t} + \mathbf{J}\]
\[ \nabla \cdot \mathbf{D} = \rho \]
\[ \nabla \cdot \mathbf{B} = 0 \]

Consider a constant $\mathbf{J}$, then $\nabla \times \mathbf{H}$ is constant and hence $\nabla \times \mathbf{E}$ is $\mathbf{0}$.

No curent, static electric field
Constant current, static magnetic field
Linearly increasing current, linearly varying magnetic field and static electric field
Quadratically increasing current, quadratically+statically varying magnetic field and linear electric field


As we can see, with polynomially varying current, $\nabla$ always vanishes. However with sinuosids this is not the case.
The oscillations ensure that the magnativ and electric fields give rise to eachother perpetually; this is the self propagating EM wave.

\begin{itemize}
reflection
creation of surface currents
\end{itemize}

THe geometry, orientation and composition of an antenna has an effect on how much it relects or allows surface currents.



\subsection{Wave equations}


From Maxwell's equtions in a charge free vacuum, one can derive the following by taking the curl of Faraday's law, applying a famous vector identity $\nabla \times ( \nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E} ) - \nabla^2 \mathbf{E}$, and using Gauss' law.

\[\nabla^2 \mathbf{E} =  \mu \varepsilon \mathbf{E}_{tt}\]



We can solve this PDE by Separation of variables.

We can also make use of Fourier analysis and solve this wavbe equation by means of taking the Fourier transform in the time domain (as opposed to space domain). This gives the solution.
\[g(\omega t - \mathbf{k}\cdot \mathbf{r})\]

Particularly, one has the following solution.
\[\mathbf{E}_0 \cos (\omega t - \mathbf{f} \cdot \mathbf{r} + \theta)\]

This will form the basis of a signal, however it is of physical interest to understand the wavevector and the direction of the wave's propagation.

Electromagnetic wave equation
Fourier analysis in solving the wave equation
cylindrical coordinates solution to wave equation 


Liearly polarized, electric field oscillates along a plane (or along some single axis)

\subsection{Propagation of radio waves}

The fact that $\mathbb{E}$ is governed by a wave equation derived from Maxwell's equations suggests the existence of electromagnetic waves by definition. Furthermore, since $c=\frac{1}{\sqrt{\mu \varepsilon}}$ (i.e lightspeed) would be the 'wavenumber' of electromagnetic waves, the suggestion arises that such waves would propagate at lightspeed; this observation led James Maxwell to believe that visible light could be a category of these electromagetic waves. These mathematical theorems were shown to correspond physically when an experiment in 1864 by Heinrich Hertz verified the theory.

Now that we are assured of the physical correspondence of our mathematical theory of electromagnetic waves, we will look into deeper properties of these waves, namely the direction and magnitude of propagation and energy. For this we admit the \emph{Lorentz force law} into our discourse.


The Poynting theorem refines our understanding of electromagnetic waves by finding a concrete form for the energy flux these waves are responsible for. This reveals the direction and magnitude of these waves as desired.

\[-u_t = \nabla \cdot \mathbf{S} + \mathbf{J} \cdot \mathbf{E}\]

The vector field $\mathbf{S}=\mathbf{E} \times \mathbf{H}$ is known as the \emph{Poynting vector}.



\subsection{Radio transmission and reception}


From the wave equation, we require . 
In essence, radio waves are generated 

How EMR is formed by AC voltage
ac voltage in antenna plus cylindrical solution to wave eq

The prime candidate for generating radio waves is by employing AC current in an antenna; current modelled by a sinusoidal function.

AC current is generated by a simple LC circuit, as illustrated below.


IMAGE OF LC CIRCUIT HERE

The fact that it produces an AC current is evident by solving the following ODE that is derived by applying \emph{Kirchoff's voltage law} along the circuit.

\[I''+\omega^2 I=0, I \in C^{\omega}\mathbb{R})\]
\[I(0)=I_0 \cos(\theta)\]
\[\omega = \frac{1}{\sqrt{LC}}\]
\emph{angular frequency}
\emph{amplitude}
\emph{phase}

Such an equation can be routinely solved by the method of characteristic equations, taking its Laplace transform and a multitude of other commonplace techniques. In all cases, the solution is the following.

\[I(t)=I_0 \cos(\omega t + \theta)\]

Furthermore, this solution offers information regarding the nature of the oscillation; one can 'tune' the AC oscillation frequency to a desired $\omega$ by ensuring that the inductance and capacitance of the circuit are set to obey $\omega = \frac{1}{\sqrt{LC}}$.


\subsection{Mathematics of antenna theory}






\section{Mathematics of signal processing}

In the late 19th century, radio waves were first considered as an engineering tool to transmit information. The idea is to oscillates electrons in the transmitting antennae to propagate radio waves in such a way that they encode information, and that these radio waves appear in some recieving antenna in the form of voltage, from which information is decoded.

The engineering technique used to encode and decode messages is called \emph{modulation}, and the voltage observed in the receiving antenna is called the \emph{signal}. 



AM - Amplitude modulation
SSB - Single side band modulation
FM - Frequency modulation


Mathematically, a modulation can be interpreted as the general form that a signal may take.
ifferent modulations have different engineering requirements due to the physical nature of the waves in question. We will and develop mathematical objects that can be used to simulate radio receiver/transmitter components as dynamical systems, ultimately creating a formal system that allowing mathematical engineers. Such a formal system is of practical interest; it would allow for engineers to generally describe a general model of a proposed radio technology with the confidence that such a model would function (provided that the engineer knows that there exist physical implementations of each dynamical system used that behave as defined).

To showcase the power of this theory for mathematical engineers, we will use these dynamical systems to construct an entirely theoretical (yet physically faithful) representation of radio devices.



\section{AM radio}

AM constitutes as a (relatively) rudimentary yet relevant modulation technique, hence this section will also introduce some universal definitions that appky to many other modulation systems.


We define a \emph{message} as the underlying function that represents the information we desire to transmit and receive; traditionally one imagines this function to represent audio information transduced by a microphone, however this really represents any type of data encodable as a mathematical function. The modulation of AM radio interprets the signal from the transmitting antennae of the following form.


\[s(t) = m(t) \cos (\omega t + \theta)\]

In AM, this sinusoid is called the \emph{carrier wave}. We also restrict $m : \mathbb{R} \to [0,\infty)$ to avoid a phenomenon called 'overmodulation', which can lead to undesired effects in envelope detection stage (more on this later). We also consider a phase $\theta$ to represent. In the most realistic analysis, $\theta$ may be a function since a moving reciever or effects from the ionosphere 


With a plethora of radio stations and physical processes linearly superimposed upon one another, it is sensible to model the voltage response in a recieving antenna by the following signal.
Recall the 
\[r(t)= n(t) + \sum^{n}_{k=1} tr_k (t) \]




\subsection{Filters}



\subsubsection{HPF and LPF}

Ideal HPF and LPF centered with cutoff frequency $\omega_c$.
\[HPF(f) = \mathcal{F}^{-1} \{ \mathcal{F}\{f\} H(\xi-\omega_c) \} \]
\[LPF(f) = \mathcal{F}^{-1} \{ \mathcal{F}\{f\} H(-\xi+\omega_c) \} \]

Since the time reversal property of the Fourier transform can be used translate many of the results for HPFs to LPFs (and vice versa), we will limit our discourse of properties to the HPF, and .

In mathematical discourse, there is no objection to representating functions in terms (unless one wants a more concrete representation perhaps).

However As an interesting sidenote, we can use the convolution theorem of the Fourier transform to give a more concrete representation of these dynamical systems. 

\[HPF(f) = f * \mathcal{F}^{-1}\{H(\xi-\omega_c)\} \]
\[HPF(f) = f * e^{ix\omega_c}[\frac{i}{2\pi t}+\frac{\delta(t)}{2}] \]



\[HPF(f) = f * [\frac{i}{2\pi t}+\frac{\delta(t)}{2}] \]
\[HPF(f) = \frac{f}{2} + \frac{i}{2\pi} (f * \frac{1}{t}) \]

This convolution is known as the Hilbert transform, and it will play an important role in SSB modulation.

 This not only gives an easier method to calculate the HPF, but interestingly it is more faithful on an actual implementation of a HPF; in practice one may not have some circuit that can take the Fourier transform of some signal, so a representation in terms of concrete functions becomes the key behind actually constructing such a circuit. Although the actual construction of such circuits is outside the scope of this thesis, this is a good opportunity to reflect on how the convolution theorem, a theoretical result, has inspired real engineering design choices.



As well as ideal HPF and LPF, a mathematical engineer may choose a to work with a more realistic mathematical model that would bring theory closer to experimentation, after all, the Paley-Wiener theorem implies limitations on possible filter quality, not to mention the difficulty in constructing filters at higher frequencies. Alas, strong approximations to ideal filters can be constructed and one can model them by convolving the signal with a more realistic function representing the effects of an actual circuit.


- CR HPF
- RL HPF
- CRCR HPF
- RCL HPF
- Active HPF


With these circuits in mind, a more faithful mathematical model for a HPF is the following.
\[HPF(f) = f * \mathcal{F}^{-1}\{\frac{\xi^2 / \omega_{c}^2}{\xi^2 / \omega_{c}^2 + \xi/ \omega_{c}Q +1}\} \]


dynamical system representing a circuit that multiplies a a spectrum by a box function (the passband). Due to the convolution theorem, it can also be interpreted as convoluting the signal by a (possibly translated) sinc function.

\[BFP(f) =  HPF \circ LPF (f)  \]
\[BFP(f) =  \mathcal{F}^{-1} \{ \mathcal{F}\{f\} B\} \]
\[BFP(f) = f * \mathcal{F}^{-1}\{B\}\]




BPFs play a major role in tuning the circuit to a specific frequency.;  are the dynamical system responsible for selecting the desired frequency, and are physically implemented by resonant LC circuits 
With a plethora of radio stations and physical processes linearly superimposed upon one another, it is sensible to model the voltage response in a recieving antenna by the following signal.
Recall that the recieving antenna must deal with the following signal. 
\[r(t)= n(t) + \sum^{n}_{k=1} m_k (t) \cos (\omega_k t - \theta) \]

Most of this signal is undesired information! The noise factor and signals from undesired radio stations are all superimposed over the one signal we wish to hone in on; a BPF is vital in removing undesired channels by admitting the desired portion of the frequency space, essentially eliminating the noise and other channels..





\subsection{Local oscillator}

A \emph{Local oscillator of frequency $\omega$} is a signal created by an AC circuit; this signal is typically mixed (mutliplied) by the received signal in the process of demodulation.

Here is the form of a local oscillator
\[LO(t) = cos(\omega t + \theta)\]

In actual implementations of radio technology, local oscillators are used to generate the carrier wave in modulation (transmitter) and are used to be mixed with a received signal to implement heterodyne demodulation (reciever). 

Imagine a scenario where a signal in the reciever's antenna has a carrier wave of the form $[m(t)+A] \cos (\omega t)$. When the reciever is turned on and the LO begins oscillating at some $t$, it is extremely likely that local oscillator is out of phase (theta is nonzero) with the received signal since it only starts oscillating when turned on; there is no reason to expect it to somehow oscillate with the desired received carrier wave.

A local oscillator that is able to always be in phase with the carrier wave is called synchronous since it is in some sence 'synced' with the carrier wave.

\subsection{Mixer}
A \emph{mixer} is a dynamical system representing a circuit that multiplies two signals. Representing an ideal mixer is a matter of simple multiplication.
\[MIX(f,g)=fg\]

In the realm of radio technology, mixers are often needed to encode a message with a carrier wave, hence one multiplicand tends to be a simple cosine function. In AM radio this is most definitely the case; the primary role of a mixer in an AM radio transmitter is to encode a message with a carrier wave (which is always a cosine wave), and a mixer with one argument as cosine also appears as a step the process radio recievers use to decode a message signal.

IMAGE OF MIXER


We now restruct our attention to such mixers where one multiplicand is fixed to cosine; their design is much simpler and hence we can create a dynamical system that models their behaviour accurately. A ring-modulator circuit can be used to represent multiplication by cosine (i.e the output of multiplying a signal and local oscillator signal AKA an AC current source)


\[MIX(f,\cos)=f\cos\]
\subsection{Envelope detector}
dynamical system representing a circuit that extracts a message riding a carrier wave. Since AM radio transmission uses the message to amplify the carrier wave, following the peaks of the signal to extract the message.

Ideally for an AM radio signal designed to avoid overmodulation, our envelope detector would be the following.
\[ED(f) = \frac{f(t)}{\cos (\omega t)}\]





An RC circuit can be used to this end, how
\[RC \cdot ED(f)' + ED(f) = f\]



\subsection{Amplifier}


An early AM radio reciever called the crystal radio required no power other than that which it extracts from the radio waves! Despite this novely, the power from the radio waves alone produces a rather weak signal; how could one possibly jam out to Van Halen on such a device?

Therefore in practice one desires an amplifier a circuit that strengthens a signal it receives. As a dynamical system, one can interpret this as a simple scaling of a signal.

\[AMP(f) = Af\]

An older circuit implementation would include a triode (vacuum tube) to this end.




\subsection{IF BPF}


Due to the impossibility of constructing ideal BPFs, generally inferior quality of dynamic BPFs (which would be a necessity if one wants to change between radio channels rather than the device being fixed to one channel), and the difficulty of constructing higher frequency BPFs, a AM receiver with only a single BPF would produce quite poor results.

Once again drawing from the power of Fourier analysis, the idea is to use cosine multiplication to shift the signal's frequency down to some fixed frequency called the \emph{intermediate frequency}, apply a BPF at this frequency, and amplify the signal. It is much easier to build quality BPFs at the intermedate frequency since not only is it a fixed frequency (hence outperforming BPFs with dynamic range), but it operates at a lower frequency to the rest of the AM radio spectrum; BPFs work better in these ranges.




we choose f_IF so that f_IF < f_LO (for any AM signal) because this implies that 4 copies (2 symmetric pairs) of the spectrum are guaranteed to not overlap (static distance 2f_{IF} between spectrum centers)

Furthermore we center our bandpass on the f_LO = f_c + f_IF pair rather than fc - f_IF  pair of the spectrum since it is easier to make LOs operating in that range  (because of a smaller frequency interval  multiplication wise).



\subsection{Limiter}

\subsection{Constructing an AM radio reciever}

We will use control theory to give a theoretical construction of an idealized and realistic \emph{superheterodyne receiver}; a popular design of AM radio receivers.

In a perfect world, one can construct the superhererodyne reciever as such.

\[ SHR(f) = AMP(BPF(MIX(f,\cos)),v)  \]

But this is not a perfect world; although this theory can be used as a sandbox for mathematicians to understand radio technology, any practical application of this theory must avoid the use of the ideal dynamical systems in favour of practical variants. In this context, the assumption of an ideal BPF limits the applicability of this model; it will be necessary model a realistic dynamic BPF and compensate by including an IF BPF.



\section{SSB radio}



AM modulation works, however there are improvements with regards to efficiency that can be made. Since our familiar AM signals are even functions (given their modulation as dynamically scaled cosine functions), their spectrum is an even function (this is a basic result of Fourier analysis).

Energy of a signal $s$ is defined as such.
\[W= \int_{0}^{\infty} s^2 (t) dt\]

By the Plancherel theorem we obtain the following alternate representation for a signal's energy in terms of the spectrum of the singal.
\[W= \frac{1}{2\ pi}\int_{\mathbb{R}} |\mathcal{F}\{s\}(\xi)|^2 d\xi\]

Recalling that our spectrum is symmetric in AM radio, finding a way to send one sideband of the signal and reflect the spectrum in the receiver's edge would halve the energy used!





\[tr(t) = m(t) \cos (\omega t) \pm \mathcal{H}\{M(t)+C\}(t) \sin(\omega t)\]



\subsection{All pass filter (APF)}


In the transmission of SSB, one trasmits a singnal containing half the spectrum, where the reciever reflects the spectrum back out to interpret the signal. As demonstrated, this requires the use of some system to conduct a Hilbert transfrom. The Hilbert transform is however non-causal, and hence much like an ideal BPF, is impossible to implement. 

All Pass filters (APFs) are a system that while conserving 'gain', shift the phase of the function by $\frac{\pi}{2}$; this models the behaviour of Hilbert transforms.

We can however build a similar aparatus to do a similar goal that concedes by assuming a finite passband and cose to 90 degree phase shift.


%\[HIL(f)=\mathcal{F}\{\frac{1}{\pi t}\}(t)f(t)\]
\[APF(f)= \mathcal{Hi}\{f\}\]

Weaver's circuit offers a realistic model for an APF,

\[APF(f)= \mathcal{Hi}\{f\}\]

Function is again based on circuit, perhaps base it on weaver's circuit for an APF.



dynamical system representing a hilbert transform (shift in 90 degrees of phase)
Allows all frequencies, however may shift them in phase. Physically implemented by weaver's circuit.


\subsection{Reception of SSB}


Let $s$ be the signal received in an SSB receiver's antenna. For simplicity, we assume it is the only signal being broadcasted (we have discussed the role of BPFs in selecting a frequency).
\[s(t) = m(t) \cos (\omega t + \theta) \pm \mathcal{H}\{m\}(t)\sin(\omega t + \theta)\]

Applying a BPF selects the following frequency

\[ \frac{m(t)}{2} \cos (\theta) \pm \mathcal{H}\{m\}(t)\sin(\omega t + \theta)\]

%\section{FM radio}



%Fourier analysis in signal multiplying and the sidebands

%a signal is multiplied by this factor so that when transmitted, it doesn't interfere with other signals
%We will focus on DSB-SC, however one can also 'mux' multiple signals, etc.
%Sampling theorem and basic chopper modulator

%fleming's envelope detector to ideal envelope detector
%fleming's envelope detector+resonant LC circuit to an ideal tunable radio
%triode to amplifier




\end{document}
