Two Birds with One Tone: I/Q Signals and Fourier Transform – Part 1

When a new member arrives at the DSP club, this is what they find at the club gate: I/Q signals. Perhaps a secret plot to keep most people out of the party?

Some return from here to try another area (e.g., machine learning, which pays more and is easier to understand but less interesting than DSP). Others persist enough to push the gate open (even a little understanding is sufficient for this task). So what exactly makes this topic so mysterious?

To investigate the answer, we start with an example audio signal drawn in the figure below that displays amplitude versus time for some spoken words.

For a comparison, a radio signal with digital modulation is plotted below.

Clearly, both audio and radio signals are simple real waveforms plotted with time. Why then I/Q processing is not common in traditional audio applications but an integral part of radio communications? For this purpose,

• Part 1 delves into the significance of I/Q processing in wireless systems, and
• Part 2 broadens the scope, linking the I/Q signals to Fourier Transform which proves advantageous in other signal applications too.

Let us start with the nature of an electromagnetic wave.

An Electromagnetic Messenger

---

In wireless systems, an electromagnetic wave acts as a messenger that carries the intended message in variations of its amplitude, phase or frequency.

The Sinusoidal Wave

---

This electromagnetic wave is produced by an oscillator through acceleration and deceleration of charges in the forward and reverse directions. Since the charges primarily oscillate back and forth, the wave exhibits a sinusoidal form.

A reverse process occurs on the receive side. The impinging EM wave influences the electrons in the antenna through its electric and magnetic fields. The acceleration and deceleration of these electrons under the influence of these fields creates a tiny sinusoidal electrical signal.

In DSP, a pure sinusoidal signal is also known as a tone. The title of this article refers to this tone, as we will now see how this simple signal alone leads us towards a deep understanding of I/Q signals now and Fourier Transform later.

Encoding the Message

---

This sinusoidal wave can be written as
\[
x(t) = A \cos (\omega t + \phi)
\]

where

• $A$ is the amplitude,
• $\omega$ is the angular frequency, and
• $\phi$ is the phase.

Any of the above three parameters can be altered with time to send a message to the receiver. At a given time, changing a parameter conveys one unit of information.

But One Carrier is Not Enough

---

In bygone times, messages were delivered either by homing pigeons traveling long distances or by spies arriving on horseback at the gates of a walled city. For obvious reasons, having more messengers was essential for surveillance and governance. Therefore, kingdoms used to rely on flocks of pigeons and networks of spies.

Similarly, the question is how to get more carriers in the case of a sinusoidal wave to deliver more messages. To achieve this goal, there should be no interference among the carriers. However, as shown in the figure below, when two carriers with different amplitudes (or phases or frequencies) combine, they create a wave with an amplitude (or phase or frequency) that differs from both of the original carriers. Clearly, two waves cannot coexist without meddling with each other.

To solve this problem, we need to learn about orthogonality.

A Numerical Prism

---

In mathematics, orthogonality is a concept that generalizes the geometric idea of perpendicularity to linear algebra.

Orthogonality

---

To understand why orthogonality is important, consider a common example of our 3D space where the vectors x, y and z are orthogonal to each other.

1. Any point in this 3D space can be written as a combination of the basis vectors with x, y and z coordinates.
   \[
   \mathbf{r} = x \mathbf{i} + y\mathbf{j}+ z\mathbf{k}
   \]

   If we find something similar in the world of signals, that can enable us to see any real-world signal as a combination of those basis signals, similar to how a prism splits the white light into seven colors.

2. No matter how far we travel in x-direction, there is absolutely no change in y and z coordinates. In a similar vein, there will be no interference among the basis signals if one of them is modified. The advantage is that they can be independently modulated with different information, whether in amplitude, phase or frequency.

As shown in the figure depicting the superposition of waves before, even smooth sinusoids are not orthogonal to each other since their superposition is not zero. However, if these sinusoids are passed through an algorithm implementing some arithmetic operations, we have a chance to achieve orthogonality at the output. This is illustrated in the figure below.

Let us call this algorithm a numerical prism. Next, we see how a simple operation can indeed act as such a prism.

The Dot Product

---

In a world of machine learning and artificial intelligence, extracting an ever increasing amount of information from the same waveform has become possible. Since DSP engineers cannot spend millions of dollars, they rely on simple mathematical operations for efficient implementation of algorithms. So what we are seeking here is a simple technique that gives us more sinusoidal waves to carry information unaffected by one another.

Let us take help from the vectors themselves. When do two vectors not interfere with each other? When the angle between them is $90^\circ$ or the cosine of the angle is zero. This implies that
\[
\mathbf{a}\cdot \mathbf{b} = |\mathbf{a}|\cdot |\mathbf{b}|\cdot \cos \theta = 0, \quad \text{or}\quad \theta = 90^\circ
\]

Just in case you are wondering where the cosine came from: In a 2D space, and without any loss of generalization, assume that one of the two vectors is $(1,0)$. In this case, the dot product is simply the projection or x-coordinate of the other vector, which brings the cosine of the angle between them into the expression.

A dot product of two length-$N$ vectors is defined as the sum of their element-by-element products.
\[
\begin{align*}
r &= \mathbf{a}\cdot\mathbf{b} = \sum_{n=0}^{N-1}a_n b_n \\\\
&= a_0b_0+ a_1b_1 + \cdots + a_{N-1}b_{N-1}
\end{align*}
\]

Vectors $\mathbf{a}=[1,0]$ and $\mathbf{b}=[0,1]$ represent a common example. But the same idea can be extended to other pairs like $\mathbf{a}=[2,3]$ and $\mathbf{b}=[6,-4]$ where we have
\[
r = \mathbf{a}\cdot\mathbf{b} = 2(6) + 3(-4) = 0
\]

In the domain of signals, this dot product is similar to correlation at zero lag defined by
\[
\begin{align*}
r &= \sum_{n=0}^{N-1} x[n]\cdot y[n]\\\\
&= x[0]y[0]+ x[1]y[1] + \cdots + x[N-1]y[N-1]
\end{align*}
\]

This process is illustrated in the figure below.

In continuous time, the summation becomes an integration and can be written as
\[
r = \int_{\tau_1}^{\tau_2} x(t)y(t)dt
\]

where the integral is taken over a defined interval.

Since a sinusoid has only three parameters, namely amplitude, phase and frequency, we now explore which of them can facilitate orthogonality.

Orthogonality in Amplitude

---

Let us start with two sinusoids stored in digital memory with different amplitudes but same frequency and phase. Then, we can write
\[
\begin{align*}
r = \int A_1\cos (\omega t + \phi)\cdot A_2 \cos (\omega t + \phi)dt &= A_1A_2\int \cos^2 (\omega t + \phi)dt \\\\
&\neq 0
\end{align*}
\]

In words, the reasoning is as follows.

• When the dot product or correlation of these two signals is computed, the amplitude can clearly be taken out as common.
• The result is a dot product of cosine with itself.
• Since a number multiplied with itself is its square, this sum of squares is a large number without any cancellation of positive values with negative ones during the integration process.
• As a consequence, it can never be zero.

We conclude that orthogonality in amplitude is not possible with a prism that computes a summation of element-wise products. Anyone who develops a different numerical prism that ensures orthogonality in amplitude is going to be famous.

Orthogonality in Phase

---

Remember that only amplitudes exist in a real world, e.g., in a computer memory, as shown in the figure below.

Can you see any phase or frequency in the signal values stored in the memory above? No, we define these parameters ourselves! This insight gives a hint that it might be possible to have a signal with different phases whose dot product is equal to zero.

Phase Shifted Sinusoids

---

For this purpose, the figure below plots a cosine with its all possible phase shifts. When a correlation or dot product is computed between this cosine and these phase-shifted versions $\cos (\omega t + \theta)$, the output is nonzero for all values except when the phase shift is $\pm 90^\circ$.

In other words, the sum of element-wise products of a cosine is zero for two versions: a negative sine and a sine. This can be seen from the following relation.
\[
\cos\left(\omega t \pm 90^\circ\right) = \cos (\omega t) \cos 90^\circ \mp \sin (\omega t) \sin 90^\circ = \mp \sin(\omega t)
\]

Since these two waves $\mp \sin(\omega t)$ are the same except for an amplitude scaling of $-1$, we can only retain one as orthogonal to cosine. Staying with the positive phase shift ($+90^\circ$) in cosine above, we choose the negative sine wave as the orthogonal signal.

Mathematically, we can verify this orthogonality as (including an immaterial factor of $2$)
\[
r = 2\int_{\tau_1}^{\tau_2} \cos (\omega t)\cdot \sin (\omega t) = \int_{\tau_1}^{\tau_2} \sin (2\omega t) = 0
\]

The result is zero under all conditions.

• If the integral is taken over an integer number of periods, the area under the curve $\sin (2\omega t)$ contains an equal number of positive and negative halves. They cancel each other out.
• When $\tau_1=-\tau_2$, any part of the signal from $0$ to $-\tau_2$ is exactly the same but with opposite sign as the part from $0$ to $\tau_2$. This is because $\sin (2\omega t)$ is an odd signal. Hence, their sum is zero.
• Even for an interval spanning a non-integer number of periods, this relation is approximately zero for large $\omega$. Why? Because a high frequency or diminishing time period implies that very little area would remain outside the balanced positive and negative halves.

  This can be seen as follows.
  \[
  \begin{align*}
  r = \int_{\tau_1}^{\tau_2} \sin (2\omega t) &= \frac{-\cos (2\omega t)}{2\omega}\Bigg|_{\tau_1}^{\tau_2} = \frac{\cos (2\omega \tau_1)\, – \cos (2\omega \tau_2)}{2\omega}\\\\
  &= \frac{\sin[\omega(\tau_2+\tau_1)]\sin[\omega(\tau_2-\tau_1)]}{\omega} \le \frac{1}{\omega} \approx 0
  \end{align*}
  \]

  The second last step follows from $|\sin(\omega t)| \le 1$. Hence, the above expression goes to zero for a sufficiently high frequency in the denominator. Indeed, carrier frequencies are high in wireless communications.

This orthogonality in phase implies that we have two carriers, not one! In addition to the original sinusoidal wave (let us assume that it is a cosine without any loss of generalization), we have another sinusoidal wave (negative sine) to send data on as a messenger, as long as we implement some version of dot product at the receiver.

Next, we explore how these two carriers can be used in a communication system.

IQ and Modulation

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The two possible options to modulate these phase orthogonal sinusoids are amplitude and frequency. Being more spectrally efficient, it is the amplitude modulation that is more widely used.

Modulation in Amplitude

---

We can vary the amplitude of the these two waves, say $I_m$ on cosine and $Q_m$ on negative sine, independent of each other (an explanation of this notation follows shortly), where $m$ is the time index for units of information ($m=0,1,2,\cdots$).

Benefiting from orthogonality, we can then combine these two streams into a single waveform before sending them into the physical medium (wireless channel, coaxial cable, etc.) at the carrier frequency $\omega$. The transmitted signal can then be written as

All this implies that an I/Q signal is a sum of an amplitude times a zero-phase cosine wave and another amplitude times a zero-phase (negative) sine wave.

Something important to note: there are no complex numbers here yet! Before we explore the role of complex numbers, we need to investigate where the terms $I$ and $Q$ come from.

What are $I$ and $Q$ Signals?

---

The phases of these two orthogonal waves need a reference with sine and cosine being the two options.

• In-phase: A cosine wave is chosen as the reference or in-phase part, from which the notation $I$ is extracted. Depending on the context, the term $I$ is used for any of the following three signals:
  • the cosine wave: $\cos (\omega t)$
  • the amplitude riding over this cosine: $I_m$
  • the amplitude-modulated cosine wave: $I_m \cos (\omega t)$

• Quadrature: In astronomy, the term quadrature refers to the position of a planet when it is $90^\circ$ from the sun as viewed from the earth. Since a negative sine wave has a phase difference of $90^\circ$ with the reference cosine wave, it is known as the quadrature part from which the notation $Q$ is extracted. Again, the $Q$ term is used for any of the following three signals:
  • the negative sine wave: $-\sin (\omega t)$
  • the amplitude riding over this negative sine: $Q_m$
  • the amplitude-modulated negative sine wave: $-Q_m \sin (\omega t)$

What follows next is an example from communication systems.

A Communication System Example

---

At the Tx side, the $I$ and $Q$ amplitude streams can be seen as two parallel wires before they are mixed with respective sinusoids and summed together to form a single signal $x(t)$ of Eq (\ref{equation-iq}) that is sent into the air. This can be seen in the block diagram below where the time index $m$ is not shown for clarity.

At the Rx side during downconversion, this single incoming signal is mixed with respective sinusoids to remove the carriers and extract $I$ and $Q$ amplitude streams. This is possible due to the orthogonality of these sinusoids and can be verified using trigonometric identities as follows.

• Multiply $x(t) $with a perfectly synchronized cosine in phase and frequency (including the scaling factor of $2$):
  \[
  \begin{align*}
  x(t) \cdot 2\cos (\omega t) &= \left[I_m \cos (\omega t) \,- Q_m \sin (\omega t)\right]\cdot 2\cos (\omega t)\\\\
  &= I_m +\underbrace{I_m \cos (2\omega t)-Q_m \sin (2\omega t)}_{\text{lowpass filtered}}\\
  &\approx I_m
  \end{align*}
  \]

  where the double frequency terms are eliminated by a lowpass filter.

• Multiply $x(t) $with negative sine (including the scaling factor of $2$):
  \[
  \begin{align*}
  -x(t) \cdot 2\sin (\omega t) &= -\left[I_m \cos (\omega t) \,- Q_m \sin (\omega t)\right]\cdot 2\sin (\omega t)\\\\
  &= Q_m\,- \underbrace{I_m \sin (2\omega t)-Q_m \cos (2\omega t)}_{\text{lowpass filtered}}\\
  &\approx Q_m
  \end{align*}
  \]

These $I$ and $Q$ streams can be seen coming out as two parallel wires on the Rx side of the above figure. This brings up the question of how to treat these two amplitudes; the options are a 2D vector and a single complex number.

Where Does the Complex Plane Come from?

---

As shown in the figure below, carrier generation and multiplication by $I$ and $Q$ streams in the frontend is a 1D signal processing operation in each arm. However, to prepare these streams for transmission on one side and demodulation of data bits at the other side, these two signals, $I$ and $Q$, need to be treated as a 2D number within the digital processing machine.

This gives rise to the following two possibilities:

• a 2D plane of vectors with matrix operations, or
• a complex plane.

While complex numbers do have some similarities with vectors in a 2D plane, they are not identical. In the 2D geometric plane, points are represented as pairs of real numbers $(x, y)$, while in the complex plane, points are represented as a single complex number $x + jy.$ Some of its implications are the following.

• Addition: Adding two 2D vectors $(x_1,y_1)$ and $(x_2,y_2)$ results in $(x_1+x_2,y_1+y_2)$, which is similar to how adding two complex numbers $x_1+jy_1$ and $x_2+jy_2$ leads to $(x_1+x_2) + j(y_1+y_2)$. Same is the case for subtraction.
• Multiplication: Multiplication is defined in complex numbers just like real numbers. For example, multiplying $x_1+jy_1$ and $x_2+jy_2$ gives the following result.
  \[
  (x_1+jy_1)(x_2+jy_2) = (x_1x_2-y_1y_2) + j(x_1y_2+y_1x_2)
  \]

  In a vector world of dot products, cross products, element-wise products and so on, multiplication is not as straightforward an operation. However, consider forming a 2D matrix from any complex number as follows.
  \[
  x_1+jy_1 \rightarrow \begin{bmatrix}x_1 & -y_1 \\ y_1 & x_1 \end{bmatrix}
  \]

  Then, the multiplication of two 2D matrices yields
  \[
  \begin{bmatrix}x_1 & -y_1 \\ y_1 & x_1 \end{bmatrix}\begin{bmatrix}x_2 & -y_2 \\ y_2 & x_2 \end{bmatrix} = \begin{bmatrix}x_1x_2-y_1y_2 & -(x_1y_2+y_1x_2) \\ x_1y_2+y_1x_2 & x_1x_2-y_1y_2 \end{bmatrix}
  \]

  Compare the first column above with the multiplication result of complex numbers. Although both give the same answer, complex numbers handle this product just like real numbers while vector processing has to go through a complicated route.

• Division: Even more problems emerge with division. In real numbers, when $x_1$ is divided by $x_2$, we can write
  \begin{equation}\label{equation-iq-signals-real-division}
  \frac{x_1}{x_2} = z \qquad \text{or}\qquad x_1 = zx_2 = x_2z
  \end{equation}

  For complex numbers, we can cancel the imaginary number $j$ in the denominator using arithmetic rules to find the result.
  \[
  \frac{x_1+jy_1}{x_2+jy_2} = \frac{x_1+jy_1}{x_2+jy_2}\cdot \frac{x_2-jy_2}{x_2-jy_2} = \frac{x_1x_2+y_1y_2+j(y_1x_2-y_2x_1)}{x_2^2+y_2^2}
  \]

  Matrix division, however, leads to two problems.

  • If the matrix $\mathbf{X_2}$ is invertible, matrix multiplication is not commutative. Analogous to division in real numbers in Eq (\ref{equation-iq-signals-real-division}), we have
    \[
    \mathbf{X_1X_2^{-1}} = \mathbf{Z} \qquad \text{or} \qquad \mathbf{X_1} = \mathbf{ZX_2} \neq \mathbf{X_2Z}
    \]

    i.e., these are two distinct matrices.

  • If the matrix $\mathbf{X_2}$ is not invertible (i.e., its determinant is 0 or it is a non-square matrix), then solutions to the above expressions either will not exist or will not be unique.

  Therefore, the concept of matrix division is not easily justified.

From the above description alone, complex numbers already appear to be more suitable for 2D signal processing operations. But Euler’s formula practically sealed the deal.

Euler’s Formula

---

In 1748, Leonhard Euler derived a formula that combined the real and imaginary parts of a complex number into a single expression.
\[
e^{j\theta} = \cos\theta+j\sin\theta
\]

This enables us to express any complex number from rectangular coordinates to polar coordinates as a single compact number.
\[
z = x+ jy = |z|e^{j\theta}
\]

where $|z|$ is the magnitude and $\theta$ is the angle of $z$.
\[
|z| = \sqrt{x^2+y^2}, \qquad \text{and}\qquad \theta = \tan^{-1}\frac{y}{x}
\]

Euler derived this result by power series expansions of the exponential and cosine/sine expressions. Most people still use that approach but I find its lack of intuitiveness unappealing.

To keep this article brief, I am providing references for an interested reader who wants to understand these topics in a better way.

• What is the origin of complex numbers and signals?
• Why does the constant $e$ appear in the representation of complex numbers (I highly recommend this one where I demonstrate how like a machine press, $e$ compresses an infinite number of rotations (disguised as multiplications) into a single quantity).
• What is the intuitive reasoning behind multiplication and division of complex numbers?

This brings us to the following conclusion.

One example of this complex connection is Quadrature Amplitude Modulation (QAM), the most widely used modulation scheme in modern communication systems.

Quadrature Amplitude Modulation (QAM)

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In Quadrature Amplitude Modulation (QAM), the two carriers are modulated in amplitude to generate the signal
\begin{equation}\label{equation-qam}
x(t) = I_m \cos (\omega t) \,- Q_m \sin (\omega t)
\end{equation}

Where do $I_m$ and $Q_m$ come from? For this purpose, in 4-QAM, for instance, the 4 modulation symbols are shown on the left side of the figure below.

Assume that the bit sequence 00101110 needs to be transmitted. There are $4=2^2$ distinct symbols which can be addressed with a pair of bits.

• In the above figure, the first two bits 00 are mapped to $\{+A,-A\}$.
• Looking at the output of the $I_m$ and $Q_m$ lookup tables for address 00, we have
  \[
  00 \quad \rightarrow \quad \{+A,-A\} \qquad \rightarrow \qquad I_m = +A, \quad Q_m = -A
  \]

• Using Eq (\ref{equation-qam}), these two amplitudes are mixed with cosine and negative sine, respectively, to form the eventual waveform as
  \begin{equation}\label{equation-qam-example}
  x(t) = +A \cos (\omega t) \,- (-A)\sin(\omega t)
  \end{equation}

Another example of this complex signal processing is a Phase Locked Loop (PLL) at the Rx side which estimates the incoming carrier phase and compensates for its rotation using complex operations.

In summary, the complex plane provides a powerful and elegant way to represent, analyze, and manipulate signals in signal processing. It offers mathematical tools and insights that are not easily accessible in the 2D real plane, making it the preferred choice for many signal processing applications.

An I/Q Signal

---

There are two main types of signals: analog and digital. The latter case is easier to understand and hence we focus on that first.

At the most elementary level, a digital I/Q signal can be described as follows.

• A digital signal is nothing but a sequence of numbers.
• A complex number is made up of two components: a real part and an imaginary part.
• A digital I/Q signal is simply a sequence of complex numbers!

An example of such a signal is drawn in the figure below.

To put it another way, this complex sequence is actually composed of two real sequences evolving with time in which one represents the real arm and the other the imaginary arm. This is similar to a complex number which is composed of two real numbers, one on real axis and the other on imaginary axis. This is shown in the figure below.

We could have easily called the I/Q signal as a complex signal. The final question is the following: In I/Q signals, where does the phase modulation enter the picture?

I/Q and Phase Modulation

---

Anyone with some information on digital modulation is familiar with Phase Shift Keying (PSK). In this kind of modulation, the phase of a sinusoid is varied according to the data bits to convey the message. For example, in QPSK, the four bit pairs 00, 01, 10 and 11 can represent four phases of a sinusoid.
\[
\begin{align*}
00 &\quad \rightarrow \quad \phi = -\frac{\pi}{4}\\
01 &\quad \rightarrow \quad \phi = -\frac{3\pi}{4}\\
10 &\quad \rightarrow \quad \phi = +\frac{\pi}{4}\\
11 &\quad \rightarrow \quad \phi = +\frac{3\pi}{4}\\
\end{align*}
\]

The resulting constellation diagram and the modulated waveform are drawn in the figure below.

Then the PSK waveform can be written as
\begin{equation}\label{equation-psk}
x(t) = A \cos (\omega t + \phi_m)
\end{equation}

where the phase $\phi_m$ depends on the data bits chosen at time $m=0,1,2,\cdots$. This brings us to a surprising contradiction.

This contradiction can be resolved as follows. Consider the trigonometric identity $\cos (\alpha+\beta)$ $=$ $\cos \alpha \cos \beta$ $-$ $\sin\alpha \sin\beta$ to open the PSK expression in Eq (\ref{equation-psk}).
\begin{equation}\label{equation-identity}
x(t) = A \cos\phi_m \cdot\cos(\omega t) \,- A\sin\phi_m\cdot \sin(\omega t)
\end{equation}

This can become the same expression as that for 4-QAM in Eq (\ref{equation-qam})
\[
x(t) = I_m\cos(\omega t)\,-Q_m\sin(\omega t)
\]

if
\begin{equation}\label{equation-inphase-quadrature}
I_m = A\cos \phi_m \qquad \text{and} \qquad Q_m = A \sin \phi_m,
\end{equation}

Using $\cos^2\phi_m+\sin^2\phi_m=1$ and $\tan\phi_m=\sin\phi_m/\cos\phi_m$, respectively, both $A$ and $\phi$ can easily be determined.
\begin{equation}\label{equation-amplitude-phase}
A = \sqrt{I_m^2+Q_m^2} \qquad \text{and}\qquad \phi_m = \tan^{-1}\frac{Q_m}{I_m}
\end{equation}

An I/Q plane illustrating such a connection is drawn in the figure below.

Plugging them back in Eq (\ref{equation-psk}) results in

In words, the combined signal also becomes a third sinusoid with a distinct amplitude and phase! This is drawn in the figure below where both amplitude and phase of the sum $I+Q$ can be observed as different from either of the constituent $I$ and $Q$ signals.

When two phase-orthogonal sinusoids are modulated in amplitude and summed together, they effectively disappear in the resultant third sinusoid. This is unlike vectors where x and y axes are still visible while drawing a resultant vector. Imagine how strange it would seem if the following relation was true.
\[
2.5x + 4.3y = -3.7x
\]

This is one reason behind the mystery of I/Q signals.

Keep in mind that this phase modulation is mostly for understanding and analysis purposes but not for implementation. Nearly all practical modems implement PSK through amplitude processing, just like a QAM signal. This becomes possible by observing the figure below where 4 modulation phases in QPSK can easily be replaced by a lookup table with corresponding $I$ and $Q$ amplitudes described before in 4-QAM.

To summarize, we send into the air a single sinusoid modulated in both amplitude and phase. In the machine, however, it is only the underlying $I$ and $Q$ amplitudes that are processed for demodulation.

Does this all mean that I/Q signals are only important from a radio communication perspective. The answer is no and there is more to it that is described in Part 2 on Fourier Transform.