【Shishuo Design】Why can’t a multiplier be used as a modulator or mixer?

Latest update time：2021-08-31 00:17

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Q: Why can't I use a multiplier as a modulator or mixer? Aren't they the same thing?

A: Not really, and it is important to understand the difference between them.

A multiplier has two analog inputs and its output is proportional to the product of the amplitudes of the two inputs (Note 1).

V _OUT = K × V _IN1 × V _IN2

Where K is a constant with dimension 1/V. In theory, a signal can be input to either input terminal and the output will not be affected.

The modulator (or mixer) also has two inputs, but the signal input is linear, and the carrier input contains a limiting amplifier, or is driven with a sufficiently large signal to be limited by it. In either case, the carrier signal becomes a square wave, so its amplitude is relatively unimportant - as long as it is large enough and its noise or amplitude variations do not appear at the output. The formula becomes:

V _OUT = K × V _SIGNAL × sgn(V _CARRIER )

Multipliers are used in analog computing. An example is calculating power in a circuit. Signals proportional to the instantaneous voltage and current are applied to the inputs of the multiplier, and its output is proportional to the instantaneous power.

Like a modulator, a multiplier encodes the amplitude of the signal input into the signal at the carrier input, but unlike a modulator, variations in the amplitude of the carrier signal also appear at its output. In communications applications using modulators, this variation is undesirable. Assuming two sinusoidal (Note 2) waves are input to a multiplier, the simplified (Note 3) formula is:

V _OUT (t) = K/2 × V _SIGNAL × V _CARRIER [cos(ω _SIGNAL +ω _CARRIER )t + cos(ω _SIGNAL –ω _CARRIER )t]

The above formula is often used in a simplified description of the modulator, but clipping the carrier signal to a square wave means that it contains odd harmonics. The simplified formula for a square wave is the odd harmonic Fourier series:

V(t) = K[cos(ωt) – 1/3cos(3ωt) + 1/5cos(5ωt) – 1/7cos(7ωt) +…]

These odd harmonics are also modulated by the carrier, so the modulator output contains not only the desired fundamental product, but also products at the odd harmonics:

V(t) = K[cos(ω _SIGNAL + ω _CARRIER )t + cos(ω _SIGNAL – ω _CARRIER )t

–1/3{cos(ω _SIGNAL + 3ω _CARRIER )t + cos(ω _SIGNAL – 3ω _CARRIER )t}

+1/5{cos(ω _SIGNAL +5ω _CARRIER )t + cos(ω _SIGNAL – 5ω _CARRIER )t}

–1/7{cos(ω _SIGNAL + 7ω _CARRIER )t + cos(ω _SIGNAL – 7ω _CARRIER )t} +…]

In many applications, these harmonic products are filtered out and ignored, but a proper description of the modulator function must include them. Sometimes they are useful, and sometimes they overlap with the fundamental product, causing unexpected results.

Therefore, before choosing a multiplier, modulator, or mixer, you should consider what your purpose is and which device will produce the least error.

Note 1: Inputs and outputs can be voltage or current, depending on the device. This example uses voltage.

Note 2: For simplicity, input and output amplitudes are normalized.

Note 3: The formula uses the cosine function which is easier to work with. The end result is the same.

Original article from Analog Devices

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【Shishuo Design】Why can’t a multiplier be used as a modulator or mixer?

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