Multipliers and Modulators

　　Although many descriptions of modulation portray it as a multiplicative process, the reality is more complicated.

　　First, for clarity, if the signal Acos(ωt) and the unmodulated carrier cos(ωt) are applied to the two inputs of an ideal multiplier, we will get a modulator. This is because the two periodic waveforms Ascos(ωst) and Accos(ωct) applied to the inputs of the multiplier (assuming a scale factor of 1 V for ease of analysis) produce an output of:

　　But in most cases, a modulator is a better circuit to perform this function. A modulator (also called a mixer when used to change frequency) is closely related to a multiplier. The output of a multiplier is the instantaneous product of its inputs. The output of a modulator is the instantaneous product of the signal at one of the modulator's inputs (called the signal input) and the sign of the signal at the other input (called the carrier input). Figure 1 shows two ways to model the modulation function: as an amplifier with positive and negative gain switched by the comparator output on the carrier input; or as a multiplier with a high-gain limiting amplifier placed between its carrier input and one of its ports. Either architecture can be used to form a modulator, but the switched amplifier architecture (used in the AD630 balanced modulator) is slower. Most high-speed IC modulators contain a translinear multiplier (based on the Gilbert cell) with a limiting amplifier in the carrier path to overdrive one of the inputs. The limiting amplifier may have high gain, allowing low-level carrier inputs—or low gain and clean limiting characteristics, requiring a relatively large carrier input to operate properly.

　　Figure 1. Two approaches to modeling modulation functions

　　We use a modulator rather than a multiplier for several reasons. Both ports of a multiplier are linear, so any noise or modulating signal on the carrier input will multiply with the signal input and reduce the output; at the same time, amplitude variations on the modulator’s carrier input are mostly negligible. Second-order characteristics can cause amplitude noise on the carrier input to affect the output, but the best modulators minimize this effect, so it is beyond the scope of this article. A simple modulator model uses a switch driven by the carrier. An (ideal) open switch has infinite resistance and zero thermal noise current, and an (ideal) closed switch has zero resistance and zero thermal noise voltage; so, although the modulator’s switches are not ideal, the modulator still has lower internal noise than a multiplier. It is also easier to design and manufacture similar high-performance, high-frequency modulators than multipliers.

　　Like an analog multiplier, a modulator multiplies two signals together; however, unlike an analog multiplier, the multiplication operation of a modulator is nonlinear. When the polarity of the carrier input is positive, the signal input is multiplied by +1; when the polarity is negative, it is multiplied by -1. In other words, the signal is multiplied by a square wave at the carrier frequency.

　　A square wave with frequency ωct can be represented using the odd harmonics of the Fourier sequence:

　　Summing this sequence: [+1, -1/3, +1/5, -1/7 + ...] gives π/4. Therefore, the value of K is 4/π, so that when a positive DC signal is applied to the carrier input, the balanced modulator can be used as a unity-gain amplifier.

　　The carrier amplitude is not important as long as it is large enough to drive the limiting amplifier; therefore, a modulator driven by the signal Ascos (ωst) and the carrier cos (ωct) produces an output that is the product of the signal and the square of the carrier:

　　This output contains the sum and difference of the frequencies of the signal and the carrier, and the signal and all odd harmonics of the carrier. In an ideal perfectly balanced modulator, there are no even harmonic products. However, in a real modulator, residual offsets at the carrier port can result in low-level even harmonic products. In many applications, a low-pass filter (LPF) can remove the higher-order harmonic product terms. Remember that cos(A) = cos(-A), so cos(ωm - Nωc)t = cos(Nωc -ωm)t, and there is no need to worry about "negative" frequencies. After filtering, the modulator output can be calculated as follows:

　　It is the same expression as the multiplier output, but the gain is slightly different. In real systems, the gain is normalized using an amplifier or attenuator, so the theoretical gain of different systems does not need to be considered here.

　　In simple applications, it is obvious that using a modulator is better than using a multiplier, but how do you define "simple"? When the modulator is used as a mixer, the signal and carrier are simple sine waves with frequencies equal to f1 and fc, respectively. The unfiltered output contains the frequency sum (f1 + fc) and frequency difference (f1 - fc), as well as the frequency sum and frequency difference of the signal and carrier odd harmonics (f1 + 3fc), (f1 - 3fc), (f1 + 5fc), (f1 - 5fc), (f1 + 7fc), (f1 - 7fc)... After LPF filtering, only the fundamental terms (f1 +fc) and (f1 -fc) are expected.

　　However, if ( f1 + fc) > ( f1 - 3fc), a simple LPF cannot be used to distinguish between the fundamental and harmonic terms, because one of the harmonic terms has a lower frequency than one of the fundamental terms. This is not a simple case, so further analysis is required.

　　If we assume that the signal consists of a single frequency f1, or if we assume that the signal is more complex and distributed in the frequency band f1 to f2, we can analyze the output spectrum of the modulator as shown in the figure below. Assuming a perfectly balanced modulator with no signal leakage, carrier leakage, or distortion, the output does not contain the input terms, carrier terms, and spurious terms. The input is shown in black (or light gray in the output figure, even though it does not actually exist).

　　Figure 2 shows the inputs—a signal in the frequency band from f1 to f2, and a carrier at frequency ffc. The multiplier excludes the following odd harmonics of the carrier: 1/3(3fc), 1/5(5fc), 1/7(7fc), …, indicated by the dashed lines. Note that the decimals 1/3, 1/5, and 1/7 represent amplitudes, not frequencies.

　　Figure 2. Input spectrum showing signal input, carrier, and odd carrier harmonics.

　　Figure 3. Output spectrum of a multiplier or modulator using an LPF

　　Figure 4. Unfiltered modulator output spectrum.

　　If the signal frequency band (f1 to f2) is within the Nyquist band (dc to fc/2), then an LPF with a cutoff frequency above 2fc will result in a modulator with the same output spectrum as the multiplier. If the signal frequency is above the Nyquist frequency, the situation is more complicated.

　　Figure 5 shows what happens when the signal band is just below fc. It is still possible to separate the harmonic and fundamental terms, but an LPF with a steep roll-off is required.

　　Figure 6 shows that since fc is within the signal passband, the harmonic terms are superimposed (3fc - f1) < (fc + f1), so the fundamental term can no longer be separated from the harmonic terms by the LPF. The desired signal must now be selected by a bandpass filter (BPF).

　　So, although modulators are superior to linear multipliers in most frequency conversion applications, their harmonic terms must be taken into account when designing practical systems.

　　Figure 5. Output spectrum when the signal is greater than fc/2

　　Figure 6. Output spectrum when the signal exceeds fc