Applied Electronics III · 120 h · Topic 3 of 10

Mixers

An FM radio tuned to 100 MHz also hears 121.4 MHz and 105.35 MHz if its input filter does not reject them. The mixer cannot tell which side the signal comes from.

Intermediate frequency Image frequency Balanced mixer Gilbert cell Spurious responses

01Translating a signal in frequency

A mixer multiplies the input signal by that of a local oscillator. By the product-of-cosines identity, the sum and the difference of the frequencies appear at the output:

\[ \cos\omega_{RF}t \cdot \cos\omega_{LO}t = \tfrac12\cos(\omega_{RF} - \omega_{LO})t \] \[ \qquad + \tfrac12\cos(\omega_{RF} + \omega_{LO})t \] A filter selects one of the two, the intermediate frequency (IF). It is the same spectrum translation as in linear modulation, used to bring the received signal down to a fixed frequency where it is easy to filter and amplify.

02The image frequency

The mixer cannot tell whether the input is above or below the local oscillator: two frequencies give the same IF. If the oscillator is above the desired signal (high-side injection), the other one is on the other side:

\[ f_{FI} = |f_{RF} - f_{LO}| \] \[ f_{imagen} = f_{RF} + 2 f_{FI} \ \ \text{(high-side injection)} \] In an FM receiver with a 10.7 MHz IF tuned to 100 MHz, the oscillator is at 110.7 MHz and the image at 121.4 MHz. A filter before the mixer has to attenuate it; that is why a higher IF makes image rejection easier, and a lower one, channel selectivity.

That trade-off leads to double-conversion receivers, with a high first IF for the image and a low second one for selectivity, and to image-reject mixers, which combine two mixers in quadrature to cancel it, as will be seen in modulators and demodulators.

03Real mixers

Type	How it works	Feature
A diode or transistor	The nonlinearity generates the products	Simple, with many unwanted products
Diode double-balanced	A ring of four diodes switched by the local oscillator	Suppresses the oscillator and the RF at the output; conversion loss of about 6 dB
Gilbert cell	Switched differential pairs, as in the MC1496	Conversion gain, integrable
Sampling or passive CMOS mixer	Switches toggled at the oscillator frequency	Good linearity in integrated receivers

The specifications of a mixer are its conversion gain or loss, the noise figure, the isolation between ports and the linearity, with the third-order intercept point from amplifier measurements. The Gilbert cell is analyzed in the MC1496 publication.

04Spurious responses

A real mixer does not only multiply: its nonlinearities generate all the products \( m f_{LO} \pm n f_{RF} \). Each combination can convert to the IF a signal that is not the desired one. An input signal \( f \) produces output at the IF if:

\[ |m f_{LO} - n f| = f_{FI} \] \[ f = \frac{m f_{LO} \pm f_{FI}}{n} \] \( m + n \) is the order of the response; the higher it is, the weaker. One of the most troublesome is the half-IF response, with m = n = 2, at \( f_{FI}/2 \) from the desired signal toward the oscillator: it is so close that the input filter barely attenuates it.

Lab · image and spurious responses of a receiver

A receiver tuned in the FM band. On the input frequency axis, from 50 to 200 MHz, the green bar is the desired signal; the red one, the image, and the gray ones, the spurious responses up to fourth order of each factor, with lower height the higher their order.

Tuned frequency =

05In the lab

Activity 1 · Conversion

Mix two generators with a double-balanced mixer and observe the output on the spectrum analyzer. Measure the conversion loss and the oscillator leakage.

Activity 2 · Image

In an FM radio, inject a signal at the image frequency and measure how much stronger it has to be than the desired one to be heard equally.

Activity 3 · Half-IF

Use a generator to find the half-IF response of the same receiver and measure its rejection.

06Common mistakes

Forgetting the image when choosing the IF. With a low IF, the image is close and the input filter does not reject it.
Driving the mixer with too little local oscillator. The diodes do not switch fully and loss and distortion increase.
Ignoring the isolation between ports. The oscillator leaking out through the antenna interferes with other receivers.
Checking only the image. Low-order spurs can fall in the band of interest.

07Self-assessment

An AM receiver with a 455 kHz IF and high-side injection is tuned to 1 MHz. Where are the oscillator and the image?

The oscillator at 1.455 MHz and the image at 1.91 MHz.

Where does the half-IF response of the FM receiver tuned to 100 MHz fall?

At 105.35 MHz, between the signal and the oscillator.

What advantage does a double-balanced mixer have?

It suppresses the local oscillator and the RF at the output, and reduces many unwanted products.

Why do modern receivers use a high IF for the first conversion?

To move the image away and be able to reject it with a fixed input filter.

08Further reading

Stephen A. Maas. Microwave Mixers. 2nd ed., Artech House, 1993. The theory and design of diode and transistor mixers.
Behzad Razavi. RF Microelectronics. 2nd ed., Prentice Hall, 2011. Integrated mixers and image-reject receiver architectures.
Wayne Tomasi. Electronic Communications Systems: Fundamentals Through Advanced. 4th ed., Prentice Hall, 2001 (Spanish edition: “Sistemas de comunicaciones electrónicas”, 4th ed., Pearson, 2003). The superheterodyne receiver, the image and the choice of the IF.