Applied Electronics III · 120 h · Topic 7 of 10

Modulators and demodulators

Half a decibel of difference between the two branches of an I/Q modulator and five degrees of quadrature error leave the image sideband barely 26 dB down. That is why transmitters are calibrated.

I/Q modulator I/Q demodulator Costas loop Image rejection Carrier leakage

01The I/Q modulator

Any modulation can be written as a complex envelope \( I(t) + jQ(t) \), and every modern modulator generates it in the same way: two mixers fed with the local oscillator in phase and in quadrature, and a summer:

\[ s(t) = I(t)\cos\omega_c t - Q(t)\sin\omega_c t \] With I and Q generated by a processor and two converters, the same hardware produces AM, FM, SSB, QPSK or QAM: only the software changes. It is the basis of the software-defined radio of the last unit.

The I/Q demodulator does the inverse operation: it multiplies the received signal by the cosine and the sine of the local oscillator and filters, and recovers I and Q. For that, the phase of the oscillator has to match that of the received carrier: it is synchronized with a Costas loop, a PLL that uses the product of I and Q as a phase detector, or it is corrected digitally after sampling.

02Classic modulators and demodulators

Modulation	Modulator	Demodulator
AM	Variation of the final amplifier's supply	Diode envelope detector
DSB and SSB	Balanced modulator; SSB by filter or by the phasing method	Product detector with local oscillator
FM	Modulated VCO, or phase modulator with the integrated message	Quadrature discriminator, PLL
Digital	I/Q modulator	I/Q demodulator with carrier and clock recovery

The fundamentals of each modulation were covered in linear modulation, exponential and digital. The phasing method for SSB is the I/Q modulator itself, with Q equal to the Hilbert transform of the message.

03Imperfections: imbalance and leakage

A real I/Q modulator has the two branches slightly different: a gain difference \( \varepsilon \) and a quadrature error \( \varphi \). With a tone, instead of a single sideband the opposite one also appears, the image, with a rejection of:

\[ IRR = \frac{1 + 2\varepsilon\cos\varphi + \varepsilon^2}{1 - 2\varepsilon\cos\varphi + \varepsilon^2} \] \( \varepsilon \) is the gain ratio of the branches (1 if they are equal). With 0.1 dB of imbalance and 1° of error, the rejection is about 40 dB; uncalibrated analog modulators are around 30 to 40 dB. A DC voltage on I or Q also produces carrier leakage, a line at the center of the spectrum.

Lab · imbalance of an I/Q modulator

On the left, the spectrum of an I/Q modulator generating a tone at \( f_c + f_m \): the image appears at \( f_c - f_m \). On the right, the constellation of a QPSK with the same imbalance: the ideal points in gray and the transmitted ones in color.

Gain imbalance =

Quadrature error =

Current transmitters and receivers measure these errors with a test signal and correct them digitally, predistorting I and Q, and reach rejections of more than 60 dB.

04In the lab

Activity 1 · Single sideband

Generate a single-sideband tone with an integrated I/Q modulator and measure with the spectrum analyzer the rejection of the opposite sideband and the carrier leakage.

Activity 2 · Correcting

Adjust by software the gain and phase of one of the branches until the image is minimized, and a DC offset to minimize the leakage.

Activity 3 · Costas loop

Simulate in Python a Costas loop that recovers the carrier of a BPSK with a frequency offset.

05Common mistakes

Assuming that the two branches are equal. An error of a few degrees limits the image rejection to 30 dB.
Forgetting the DC offset of the converters. It shows up as carrier leakage.
Demodulating without recovering the phase. The constellation rotates and the symbols get confused.
Confusing the modulator's image with the receiver mixer's image. They have the same origin, but appear in different places.

06Self-assessment

What image rejection does a modulator with no gain imbalance and 1° of quadrature error give?

About 41 dB.

What signal has to be put on Q to generate SSB with the phasing method?

The Hilbert transform of the message.

What does a Costas loop do?

It recovers the carrier phase from the modulated signal, using the product of I and Q as a phase detector.

What does a DC offset on the I branch produce?

Carrier leakage: a line at f_c.

07Further reading

Behzad Razavi. RF Microelectronics. 2nd ed., Prentice Hall, 2011. Quadrature modulators and demodulators and the effect of their imperfections.
Qizheng Gu. RF System Design of Transceivers for Wireless Communications. Springer, 2005. System design of transmitters and receivers, with the imbalance budgets.
Wayne Tomasi. Electronic Communications Systems: Fundamentals Through Advanced. 4th ed., Prentice Hall, 2001 (Spanish edition: “Sistemas de comunicaciones electrónicas”, 4th ed., Pearson, 2003). Analog modulators and demodulators from the circuits up.