SDR_Tutor/IQ_variables.html at master · lanceman2/SDR_Tutor · GitHub

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    <h5><a href="index.html">table of contents</a></h5>

    <h1>IQ variables</h1>

    <p>
    Warning: variable names in this may not be standard.  I don't have
    access to what the standard is.  I don't even know if there is a
    standard.  It seems to vary in all the references I have access to.
    Show me what you'd call a standard, and I'll tell you how it's not
    self consistent.  Nevertheless, I needed to define a standard for
    myself.  Here it is.
    </p>

    <p>
    At some point in a circuit in that is feeding our SDR (software define
    radio) we say there is a electric potential that varies continuously
    with time.  This point is somewhere between an AtoD (analog to
    digital) (DtoA for tx) converter and an antenna.  We'll call this
    electric potential the "wave signal".  If the power flow in this
    circuit at this point is toward the antenna than we have a transmitter
    (tx), and if the power in this circuit at this point is away from
    the antenna than we have a receiver (rx).  We call this electric
    potential the wave signal.  Following most engineering conventions we
    drop the part of the model that this is an electric potential, and
    consider it to be just a real number that varies in time, without
    units.  We consider it to be some kind of abstract signal wave form,
    signal for short.  We need this signal, which is just a function of
    time, \(t\), (with no physical units), in order to define what
    \(I(t)\) and \(Q(t)\) are.
    </p>

    <p>
    Here we present an attempt to define \(I\) and \(Q\) in terms of a
    predefined wave signal.  In doing this we decide on some sign and
    phase conventions as to how the signal wave is defined, that are
    otherwise arbitrary.  We don't have access to standards that
    would clear up this arbitrariness, so we just make it up here.
    </p>

    <p>
    We define a continuous signal \(x(t)\) with

    $$x(t) = A(t) \cos(2 \pi f t + \phi(t)) \label{x}$$

    where \(A(t)\) we call the amplitude as a function of time, \(t\), the
    constant \(f\) we call the carrier frequency, and \(\phi(t)\) we call
    the applied phase shift as a function of time, \(t\).
    Note that in the case where the signal is just a single tone \(A(t)\)
    would be a nonzero constant and \(\phi(t)\) could be any finite
    constant.
    </p>

    <p>
    The choose of the phase dependence of the signal seems arbitrary.  We
    could have picked the case that when the value of \(x(t)\), in
    \ref{x}, with \(\phi(t) \to 0\), we have a sine wave, a cosine wave,
    or in general a sinusoidal that is phase shifted by any constant.
    Convention clearly shows that it is a cosine function, and hence
    \(I(t)\) is "in-phase" with it, as they say; as we will see below.
    </p>

    <p>
    There is also the question of whither of we should have \(+ \phi(t)\)
    or have that replaced by \(- \phi(t)\).  The references that I have
    found do not make clear argument as to which is more common.
    I may have to change this in the future.
    </p>

    <p>
    It's important to note that the form of this function, in a sense,
    defines what a "carrier wave" is, the \(A(t)\) and \(\phi(t)\) are
    unspecified so \(x(t)\) can be any function of time, \(t\), not just
    something close to a sine wave.  In most applications it is close to a
    simple sinusoidal wave.  Later we will combine many signal waves each
    with a different carrier frequency, \(f\).
    </p>

    <p>
    For now we start with just one "signal", by definition, if you will.
    From there will derive an ODE (ordinary differential equation) and
    generalize this ODE; to add ideas like noise to the signal, for
    example.  Giving this kinematic form, \ref{x}, for the model of a
    wave signal limits what we can do.  This ODE model, that we refer to,
    could be thought of as a reduction from solving Maxwell's equations,
    but we, in a sense, will just imperially derive these ODEs.  More on
    that later.
    </p>

    <p>
    We make a change of dynamical variables from \(A(t)\) and \(\phi(t)\)
    to new dynamical variables \(I(t)\) and \(Q(t)\) defined by

    $$x(t) =
    I(t) \cos(2 \pi f t) + Q(t) \sin(2 \pi f t) \label{IQ} .$$

    It's important to note this is a time dependent change of variables.
    Convention says that \(I(t) \cos(2 \pi t)\) is called the in-phase
    part of the signal, and \(Q(t) \sin(2 \pi f t)\) is the out-of-phase
    part of the signal.  That makes sense given that with \(Q\) and
    \(\phi\) both set to \(0\), and \(I\) held constant the signal is \(I
    \cos(2 \pi f t)\) where \(I\) is \(A\).  I'm pretty sure this,
    equation \ref{IQ}, is standard.  The alternative is \(x(t) = I(t)
    \cos(2 \pi f t) - Q(t) \sin(2 \pi f t)\).  I have never seen it
    expressed that way.
    </p>

    <p>
    Euler's formula is

    $$e^{i \theta} = \cos(\theta) + i \sin(\theta) \label{e}$$

    where \(e\) is Euler's number, \(i\) is \(\sqrt{-1}\), and
    \(\theta\) is any complex number.

    With \ref{e} and
    $$e^{i (x + y)} = e^{i x} e^{i y}$$
    we get
    $$\cos(x + y) + i \sin(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y) +
    i \left[\cos(x) \sin(y) + \sin(x) \cos(y) \right] \label{ee}$$
    where \(x\) and \(y\) are any real numbers.
    The imaginary and real parts of \ref{ee} give the two trig
    identity equations
    $$\sin(x + y) = \cos(x) \sin(y) + \sin(x) \cos(y) \label{sinsum}$$
    and
    $$\cos(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y) .\label{cossum}$$
    </p>

    <p>
    Using \ref{cossum}, with \(x \to \phi(t)\) and \(y \to 2 \pi f t\)
    we can rewrite \ref{x} as

    $$x(t) =
    A(t) \cos(\phi(t)) \cos(2 \pi f t) -
    A(t) \sin(\phi(t)) \sin(2 \pi f t) . \label{IQsum}$$

    By comparing \ref{IQ} and \ref{IQsum} we see that we can also define
    \(I(t)\) and \(Q(t)\) by
    $$I(t) = A(t) \cos(\phi(t)) \label{IQ_I} $$
    and
    $$Q(t) = - A(t) \sin(\phi(t)) . \label{IQ_Q}$$
    </p>

    <p>
    We can define carrier phase \(\theta(t)\) by

    $$\theta(t) \equiv 2 \pi f t . $$

    Now we see that \(I\) and \(Q\) are independent of the carrier phase,
    \(\theta\).  By using \(I\) and \(Q\) as
    our dynamical variables we do not concern our selves with the
    signal "carrier".  To get our original signal \(x(t)\) back
    from, this magical change of variables, \(I(t)\) and \(Q(t)\)
    we need the carrier frequency.
    </p>

    <p>
    Note: if we are not given the carrier frequency, \(f\), and are just
    given \(I(t)\) and \(Q(t)\) as a function of \(t\) we can say we still
    have the signal, but with no signal carrier.  It's like removing the
    carrier from the signal.
    </p>

    <p>
    Summing the squares of equations \ref{IQ_Q} and \ref{IQ_I}, and given
    the Pythagorean trigonometric identity

    $$1 = \cos^{2}(x) + \sin^{2}(x)$$

    we can show that

    $$A^{2}(t) = I^{2}(t) + Q^{2}(t) .$$

    Dividing equations \ref{IQ_Q} and \ref{IQ_I}, and given the
    trigonometric relation

    $$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

    we can show that

    $$\tan(\phi(t)) =  - \frac{Q(t)}{I(t)} \label{tanIQ}.$$
    </p>


    <h3>External References</h3>

    <ul>
        <li>
            <a href="https://en.wikipedia.org/wiki/In-phase_and_quadrature_components"
                >wikipedia IQ</a> As of 2024, Jan 16, they are using the
            same phase and sign conventions in signal and IQ definitions
            that I use here.  Had I seen this before I started writing
            this, I may not have written this page.  I did not write the
            wikipedia page myself because I do not consider myself an
            authority.  The point of writing this page was for my own
            reference.
        </li>
        <li><a
            href="https://www.youtube.com/playlist?list=PLu0BPYzTjiHru1KmPThmbY-8rRm3EWvUQ"
            >Software Defined Radio with HackRF by Michael Ossmann</a>
        </li>
        <li><a href="https://pysdr.org/content/sampling.html">A Guide to
                SDR and DSP using Python</a>
        </li>
        <li>YouTube video lecture: <a
                href="https://www.youtube.com/watch?v=h_7d-m1ehoY">Basics
                of IQ Signals and IQ modulation & demodulation - A
                tutorial</a>
        </li>
        <li>Science Direct: <a
            href="https://www.sciencedirect.com/topics/engineering/carrier-phase"
            >carrier-phase</a>
        </li>
    </ul>


    <h5><a href="index.html">table of contents</a></h5>

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