Modulation is fundamental to electronic communications. The modulating signal might be analog in nature (voice or music) or digital bit streams. Most modern communication systems are digital, using discrete levels of amplitude or phase to represent the data being transmitted. The more unique conditions that can reliably travel from a transmitter to a receiver, the more data you can send in a given period of time. Quadrature modulation is widely used in digital communications systems up to and including 5G.

The basic idea behind modulation is to control one or more parameters of an RF carrier by the modulating signal. Mathematically, we can express this as follows:



Where:
a(t) is the amplitude modulation (AM) term
Θ(t) is the phase modulation (PM) term
fc is the carrier frequency

This signal has its amplitude controlled by a(t) and its phase controlled by Θ(t). To implement amplitude modulation (AM), we’d apply the modulating signal as a(t) and set Θ(t) to zero. Similarly, a phase-modulated (PM) signal would have a(t) set to a constant and the modulating signal would be applied to Θ(t). For now, we’ll ignore frequency modulation (FM) but we will show that FM can be created using PM.

Vector representation

The vector representation is a convenient way of representing a modulated signal by defining inphase (I) and quadrature (Q) components.

Using the trig identity:



We can express the modulated signal in this form:

which can be reformed to pull out the I and Q components:



where





Figure 1 shows this graphically, with the I component on the horizontal axis while the Q component is on the vertical axis. This format should look familiar to EEs and is based on the 90 degrees of phase offset between the sine and cosine functions.

Figure 1 The vector diagram represents the amplitude and phase of a modulated signal. [Adapted from Ref 1]

The amplitude and phase of the modulated signal are related to the I and Q components by these equations:



I kept the “(t)” in the equations to emphasis that these variables are a function of time and will normally be changing based on the applied modulation. For classic AM, imagine the vector changing in length (amplitude) while the phase angle remains constant. For PM, imagine the opposite: the amplitude of the vector remains constant but the angle changes accordingly to the modulation. Squint real hard and you can see the vector moving around in real time.

Now this may seem like just a bunch of trig trickery, but quadrature modulation is normally implemented in a system with a block diagram shown in Figure 2.

quadrature modulatorFigure 2 A quadrature modulator uses sine and cosign functions to modulate the oscillator's carrier.

We can think of i(t) as controlling the inphase (cosine) portion and q(t) controlling the quadrature (sine) portion. Summing these together creates the desired output signal. This block diagram can be implemented using either analog or digital techniques (or a combination of both). Practical systems have been created using both approaches but, no surprise, the clear trend is to use digital circuits and digital signal processing.

Fig. 2 represents the transmit side of a quadrature modulation system. At the receive end, there will be a corresponding quadrature detector that extracts the I and Q signals from the modulated waveform.

Digital modulation

Quadrature modulation can be used to implement an endless number of modulation schemes, but it has the most value for digital modulation. For example, digital modulation using the phase of the vector is referred to as Phase Shift Keying (PSK).

Figure 3 shows two examples of PSK: 4-PSK uses four different phases to produce four modulation states. (Note that the amplitude remains constant.) Figure 3 only shows where the tip of the vector would land, which is a common way to display these states. This type of figure is often referred to as a constellation diagram. Because the modulation format has four possible states, each modulation state can represent two binary values (shown as 00, 01, 10, 11 in the diagram).

Figure 3 Constellation diagrams for simple PSK signals. [From Ref 3]

Fig. 3 also shows 8-PSK which uses phase modulation to create eight modulation states. Three-bits worth of logical states are shown on the diagram. Having more modulation states lets a system transmit more bits of information in a given time (at the expense of increased error rate in the presence of noise).

Quadrature amplitude modulation (QAM) uses both amplitude and phase to add additional modulation states. Figure 4 shows 16-QAM (with 16 states). Imagine our modulation vector jumping around, pointing to each one of these states based on the digital modulation. The logical values are not shown in the figure due to keep it simple, but the modulation states are mapped to sixteen values, representing four bits of information.

Figure 4 Constellation diagram for a 16-QAM signal. [From Ref 3]

What about FM?

As you can see, modulating the amplitude and phase of the carrier is a flexible way to create a modulated carrier. Even though it is an old technology from the 1920s, FM is still used today in applications such as broadcast and land-mobile radio. How do we implement FM using quadrature modulation?

In general, instantaneous frequency is the derivative of instantaneous phase [Ref 4].



Where f(t) is the instantaneous frequency and θ(t) is the instantaneous phase

For FM, the instantaneous frequency must change based on the modulating signal.



where kd is the deviation constant and m(t) is the modulating signal.

Solving for the desired phase signal, we get:



This result shows that we can create an FM signal by providing phase modulation that is the integral of the modulating signal. (Yes, I ignored the initial condition of the integral.)

The required PM signal can be created using an analog integrator or the equivalent digital algorithm. Thus, a quadrature modulator can produce an FM signal using PM.

Quadrature modulation and I/Q signals are broadly used in electronic communication systems. In particular, digital modulation makes good use of a quadrature modulation system. However, any carrier-based modulation can be produced, including legacy modulations types such as AM and FM. The concept of I and Q digital streams is used in many electronic communication systems due to its flexibility and has become a de facto standard for representing modulated signals.

References

  1. Spectrum and Network Measurements (2nd Edition), Section 6.12 Quadrature Modulation, Robert A. Witte, SciTech Publishing, 2014.
  2. Digital Modulation in Communications Systems- An Introduction,” Application Note, Publication Number 5965-7160E, Keysight Technologies, 2014.
  3. Modulation Schemes: Moving Digital Data With Analog Signals,” Andrew W. Davis, EE Times, 4 Oct 1997.
  4. Instantaneous Phase and Frequency,” Wikipedia.


Bob Witte is President of Signal Blue LLC, a technology consulting company.