Communication Systems · 96 h · Topic 3 of 11

The complex envelope

A 100 MHz radio signal can be simulated entirely with two numbers that change a few thousand times per second. The carrier carries no information: it only moves it.

Bandpass signal I and Q components Complex envelope I/Q receiver PAPR

01A radio signal is a carrier with two pieces of data

Almost all communication signals are bandpass: they occupy a narrow band around a carrier of frequency \( f_c \) much greater than their bandwidth. An FM radio occupies 200 kHz around 100 MHz; a Wi-Fi channel, 20 MHz around 2.4 GHz. Any such signal can be written as a carrier whose amplitude and phase change slowly:

\[ x(t) = a(t)\cos\big(2\pi f_c t + \varphi(t)\big) \] \[ x(t) = x_I(t)\cos 2\pi f_c t - x_Q(t)\sin 2\pi f_c t \] \( a(t) \) is the envelope and \( \varphi(t) \) the phase; \( x_I \) and \( x_Q \), the in-phase and quadrature components. All the information is in these slow functions; the carrier only shifts it in frequency.

02The complex envelope

The two components are combined into a single complex number that varies in time, the complex envelope:

\[ g(t) = x_I(t) + j\,x_Q(t) = a(t)\,e^{j\varphi(t)} \] \[ x(t) = \operatorname{Re}\left\{ g(t)\, e^{j2\pi f_c t} \right\} \] The bandpass signal is the real part of the envelope multiplied by the complex carrier. In frequency, the spectrum of \( x \) is that of \( g \) shifted to \( \pm f_c \).

The practical consequence is enormous: everything that happens to a radio signal —modulation, filtering, noise, demodulation— can be analyzed and simulated with its complex envelope, at low frequencies, without dealing with the carrier. This is how simulators and software-defined receivers work.

03Every modulation is an envelope

Modulation	Complex envelope \( g(t) \)	In the I/Q plane
AM	\( A_c\,[1 + m(t)] \)	Moves along the real axis
DSB-SC	\( A_c\, m(t) \)	Along the real axis, crossing the origin
SSB	\( A_c\,[m(t) + j\,\hat m(t)] \)	Rotates in one direction only
PM	\( A_c\, e^{j k_p m(t)} \)	On a circle
FM	\( A_c\, e^{j k_f \int m\, dt} \)	On a circle
QAM and PSK	\( \sum_k (a_k + j b_k)\, p(t - kT) \)	Jumps between points of a constellation

\( \hat m(t) \) is the Hilbert transform of the message. Angle modulations have a constant envelope: \( |g| \) does not change, and the information is all in the phase.

Lab · the complex envelope in the I/Q plane

On the left, the trajectory of \( g(t) \) in the I/Q plane; on the right, a few cycles of the bandpass signal \( x(t) \) with its envelope. The message is a sinusoid (or random symbols, in QPSK).

Index =

04The I/Q receiver

To recover the complex envelope, the receiver multiplies the signal by \( \cos 2\pi f_c t \) and by \( -\sin 2\pi f_c t \) and filters the two products with lowpass filters: it obtains \( x_I/2 \) and \( x_Q/2 \). This is the quadrature demodulator, and with two converters behind it, it is the front end of every modern digital receiver: once \( g(t) \) is in samples, the demodulation of any modulation is software. The converters at that input are those of AD and DA converters.

05Envelope and amplifiers

The ratio of peak power to average power (PAPR) of the envelope decides what amplifier is needed. With a constant envelope, like FM, a saturated class C amplifier will do, with an efficiency of 70 % or more. With a variable envelope, like AM or QAM, a linear amplifier operating below its maximum is needed: otherwise it clips the peaks, distorts and widens the spectrum. A 4G channel, with hundreds of subcarriers, has a PAPR of 10 dB or more, and its amplifier is one of the big energy consumers of a base station.

06In the lab

Activity 1 · A software-defined receiver

With a low-cost SDR receiver, capture I/Q samples of an FM radio and plot their trajectory in the complex plane.

Activity 2 · Quadrature demodulator

Build an I/Q demodulator for a 455 kHz AM signal with two mixers and a 90° phase shifter, and view the two outputs on the oscilloscope in X-Y mode.

Activity 3 · PAPR

With an oscilloscope, measure the peak and average power of a 100 % AM signal and compare them with 4.26 dB.

07Common mistakes

Forgetting that the envelope is complex. A single component loses the phase.
Confusing the real envelope with the complex one. The real one is the magnitude \( |g| \).
Using a saturated amplifier with a variable-envelope modulation. It distorts and pollutes the neighboring channels.
Forgetting the imbalance between I and Q. A gain or phase error between the two branches creates an image of the spectrum.

08Self-assessment

What is the complex envelope of \( x(t) = 3\cos(2\pi f_c t + \pi/4) \)?

\( g = 3e^{j\pi/4} \), constant.

What PAPR does an AM signal with index 1 and a sinusoidal message have?

\( 4/1.5 \): 4.26 dB.

Why can FM be amplified with a saturated amplifier?

Because its envelope is constant: the information is in the phase, which saturation does not alter.

What two multiplications does a quadrature demodulator perform?

By \( \cos 2\pi f_c t \) and by \( -\sin 2\pi f_c t \), followed by lowpass filters.

09Further reading

Leon W. Couch. Digital and Analog Communication Systems. 8th ed., Pearson, 2013. Uses the complex envelope as the central tool for all modulations.
Richard G. Lyons. Understanding Digital Signal Processing. 3rd ed., Prentice Hall, 2010. The quadrature signal explained intuitively, with the I/Q receiver.
Simon Haykin and Michael Moher. Communication Systems. 5th ed., Wiley, 2009. The representation of bandpass signals and their spectrum.

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