Take you deeper into the characteristics and applications of the modulator

Despite common descriptions suggesting that modulation is a straightforward multiplication process, the reality is more intricate. While many illustrations depict modulation as a multiplication operation, the actual scenario involves additional complexities. For instance, if we apply a signal Acos and an unmodulated carrier cos(ωt) to the two inputs of an ideal multiplier, we essentially create a modulator. Assuming both Ascos(ωst) and Accos(ωct) are fed into the multiplier’s inputs (with a voltage scale factor of 1 V for simplification), the resulting output becomes: Vo(t) = ½AsAc[cos((ωs + ωc)t) + cos((ωs – ωc)t))]. If the carrier Accos(ωct) has an amplitude of 1 V (Ac = 1), the equation simplifies further: Vo(t) = ½As[cos((ωs + ωc)t) + cos((ωs – ωc)t)]. In most practical scenarios, a modulator proves to be a more efficient solution for this purpose. A modulator, also referred to as a mixer when altering frequency, is closely related to a multiplier. The output of a multiplier is the instantaneous product of its inputs. On the other hand, the output of a modulator is formed by multiplying one input (the signal input) by another (the carrier input) which alternates between +1 and -1 based on the carrier's polarity. This effectively means the signal is multiplied by a square wave at the carrier frequency. Figure 1 illustrates two ways to model the modulation function: 1. Used as an amplifier to toggle between positive and negative gains based on the comparator output of the carrier input. 2. Utilized as a multiplier with a high-gain limiting amplifier placed between the carrier input and one of its ports. Both these architectures can be employed to construct a modulator. However, the switching amplifier architecture, such as that used in the AD630 balanced modulator, operates at a slower pace. High-speed IC modulators typically feature a translinear multiplier (often based on Gilbert cells) and include a limiting amplifier on the carrier path to boost one of the inputs. These limiting amplifiers might offer high gain, enabling low-level carrier inputs, or they could provide low gain with clean limiting properties, necessitating larger carrier inputs for optimal performance. We often prefer a modulator over a multiplier due to several reasons. Both ports of a multiplier are linear, so any noise or modulated signal from the carrier input gets multiplied by the signal input, thereby reducing the output. Conversely, the amplitude variations of the modulator's carrier input can usually be disregarded. The second-order characteristic causes the amplitude noise of the carrier input to influence the output, but the best modulators minimize this effect—a topic beyond the scope of this discussion. A simple modulator model employs a carrier-driven switch. An ideal open-circuit switch exhibits infinite resistance and zero thermal noise current, whereas an ideal closed-circuit switch shows zero resistance and zero thermal noise. Even though a modulator's switching isn't perfect, it still generates less internal noise compared to a multiplier. Moreover, a high-performance, high-frequency modulator is easier to design and manufacture than a multiplier. Similar to an analog multiplier, a modulator multiplies two signals. Unlike a multiplier, however, the multiplication performed by a modulator is non-linear. When the carrier input's polarity is positive, the signal input is multiplied by +1; when negative, it is multiplied by -1. In essence, the signal is multiplied by a square wave at the carrier frequency. A square wave with a frequency of ωct can be expressed via the odd harmonics of the Fourier series: K[cos(ωct) – 1/3cos(3ωct) + 1/5cos(5ωct) – 1/7cos(7ωct) + ...], where the summation of the sequence [+1, –1/3, +1/5, –1/7 + ...] equals π/4. Thus, the K value becomes 4/π, ensuring that applying a positive DC signal to the carrier input turns the balanced modulator into a unity gain amplifier. The carrier amplitude isn't critical, as long as it's sufficient to drive the limiting amplifier. Consequently, the output produced by a modulator driven by the signal Ascos(ωst) and the carrier cos(ωct) is the product of the signal and the square of the carrier: 2As/π[cos(ωs + ωc)t + cos(ωs – ωc)t – 1/3{cos(ωs + 3ωc)t + cos(ωs – 3ωc)t} + 1/5{cos(ωs + 5ωc)t + cos(ωs – 5ωc)t} – 1/7{cos(ωs + 7ωc)t + cos(ωs – 7ωc)t} + ...]. The output comprises the sum and difference frequencies of the following terms: all odd harmonics of the signal to the carrier, signal and carrier. There are no even harmonic products in an ideal balanced modulator. However, in real-world modulators, residual carrier offsets result in low-order even harmonic products. In numerous applications, a low-pass filter (LPF) eliminates higher harmonic product terms. Recall that cos(A) = cos(–A), so cos(ωm – Nωc)t = cos(Nωc – ωm)t, eliminating concerns about "negative" frequencies. After filtering, the modulator output simplifies to: 2As/π[cos(ωs + ωc)t + cos(ωs – ωc)t], which mirrors the expression output by the multiplier, albeit with a slightly different gain. In practical systems, the gain is normalized using an amplifier or attenuator, making theoretical gain differences irrelevant. In simpler applications, using a modulator clearly outperforms a multiplier. But how do we define "simple"? When the modulator serves as a mixer, the signal and carrier inputs are simple sine waves with frequencies f1 and fc, respectively. The unfiltered output contains the sum (f1 + fc) and difference (f1 – fc) frequencies, along with odd harmonic products of the signal and carrier (f1 + 3fc), (f1 – 3fc), (f1 + 5fc), (f1 – 5fc), (f1 + 7fc), (f1 – 7fc). Post-LPF filtering, the goal is to retain only the fundamental terms (f1 + fc) and (f1 – fc). Yet, if (f1 + fc) > (f1 – 3fc), a basic LPF cannot distinguish between the fundamental and harmonic terms because the frequency of a harmonic term falls below a specific fundamental term. This represents a more complex case, requiring deeper analysis. Assuming the signal contains a single frequency f1, or if it’s more complex and spans frequency bands f1 to f2, we can examine the modulator's output spectrum, as depicted in the subsequent figures. Assuming a perfectly balanced modulator with no signal leakage, carrier leakage, or distortion, the output contains no input terms, carrier terms, or spurious terms. The input appears in black (or light gray in the output image, though it technically doesn’t exist). Figure 2 presents the inputs: signals in the f1 to f2 band and a carrier at frequency fc. The multiplier lacks the odd carrier harmonics: 1/3 (3fc), 1/5 (5fc), 1/7 (7fc)..., indicated by dashed lines. Note that the fractions 1/3, 1/5, and 1/7 denote amplitude, not frequency. Figure 3 displays the multiplier or modulator output and the LPF with a cutoff frequency of 2fc. Figure 4 shows the unfiltered modulator output (excluding harmonic terms above 7fc). If the signal bands f1 to f2 lie within the Nyquist band (DC to fc/2), an LPF with a cutoff frequency exceeding 2fc ensures the modulator produces the same output spectrum as the multiplier. However, if the signal frequency surpasses the Nyquist frequency, the situation becomes more complex. Figure 5 demonstrates what occurs when the signal band is just below fc. Separating the harmonic term from the fundamental term remains possible, but this requires an LPF with steep roll-off characteristics. Figure 6 reveals that since fc resides within the signal passband, the harmonic term overlaps (3fc – f1) < (fc + f1), making it impossible to isolate the fundamental term from the harmonic term via an LPF. Now, the desired signal must be selected by a bandpass filter (BPF). Thus, while modulators excel over linear multipliers in most frequency conversion applications, their harmonics must always be factored into the system design.

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