Demodulating Double-Sideband AM Signals

So far in this series of articles, we’ve explored two forms of double-sideband amplitude modulation (AM) and their associated modulation circuits. Modulation, as we know, is the process of translating a baseband message signal to an RF frequency band for transmission. But how do we recover the message from the modulated signal once it’s received?

In this article, we’ll turn our attention to the question of demodulation. For the most part, we’ll focus on double-sideband suppressed-carrier (DSB-SC) signals. At the end of the article, however, we’ll also discuss a double-sideband demodulation circuit that transmits a carrier component.

 

DSB-SC Modulated Signals

Before we discuss demodulation, let’s briefly review what we know about DSB-SC modulation. To obtain a DSB-SC signal, we use a carrier wave with the following form:

$$c(t)~=~A_c\cos(\omega_ct)$$

Equation 1.

 

where:

A_c = Carrier amplitude

ω_c = Carrier frequency (in rad/s)

t = Time.

We then multiply the baseband message signal (m(t)) by the carrier wave, producing:

$$s(t) ~=~ m(t) ~\times~ A_c\cos(\omega_c t)$$

Equation 2.

 

In the frequency domain, this multiplication corresponds to a convolution of the baseband signal's spectrum (M(f)) with the spectrum of the cosine function. These spectrums are denoted in Figure 1 by M(f) and C(f), respectively.

 

Figure 1. Multiplication in the time domain corresponds to a convolution of the baseband spectrum with the carrier in the frequency domain (top). This translates the baseband spectrum by ±f_c (bottom).

 

In the bottom half of Figure 1, we see that the spectrum of the modulated wave (S(f)) has two copies of the baseband spectrum: one shifted to the carrier frequency (f_c) and the other shifted to –f_c.

 

Basic DSB-SC Demodulation

Given an ideal channel—one free from noise and distortion—the received signal is identical to the transmitted DSB-SC signal:

$$r(t)~=~ s(t) ~=~ m(t) ~\times~ A_c \cos(\omega_c t)$$

Equation 3.

 

To demodulate the signal, the receiver must generate a carrier with the same frequency and phase as the original carrier. This is referred to as coherent or synchronous demodulation. We then multiply r(t) by the receiver’s carrier wave and apply a low-pass filter with a suitable bandwidth. Figure 2 illustrates the demodulation process.

 

Figure 2. Demodulation of the DSB-SC signal.

 

Assume that the locally generated carrier in the receiver has a phase error of ϕ with respect to the original carrier wave:

$$c_{r}(t) ~=~ \cos(\omega_c t ~+~ \phi)$$

Equation 4.

 

The signal at the output of the multiplier is:

$$\begin{eqnarray}s_A(t) &~=~& c_{r}(t)r(t) ~=~ \cos(\omega_c t ~+~ \phi) ~\times~ m(t) ~\times~ A_c \cos(\omega_c t) \\&~=~& \frac{1}{2}A_c m(t) \Big [ \cos(\phi) ~+~\cos(2 \omega_c t ~+~ \phi) \Big ] \end{eqnarray}$$

Equation 5.

 

The first term restores the baseband spectrum. The second term produces replicas of the baseband spectrum centered at twice the carrier frequency. Figure 3 shows the spectrum we obtain after multiplying the received signal by the local carrier (assuming that the modulated signal spectrum is as shown in Figure 1).

 

Figure 3. The spectrum of the signal at the output of the multiplier (node A in the demodulator diagram).

 

Since the message signal’s bandwidth (B) is much lower than the carrier frequency (f_c), we can use a low-pass filter to suppress the signal components centered at 2f_c. This leaves us with the baseband spectrum at the output:

$$s_{out}(t) ~=~ \frac{1}{2}A_c m(t) \cos(\phi)$$

Equation 6.

 

Equation 6 shows that the output spectrum is influenced by the phase error between the carrier wave used at the transmitter and that generated at the receiver. For a non-zero ϕ, the amplitude of the output signal is reduced by a factor of cos(ϕ). For instance, if ϕ = 45 degrees, the amplitude of the output signal is decreased by a factor of about 0.7 and the output power is reduced by half. When ϕ = 90 degrees, the output signal reduces to zero.

If the phase error remains constant during signal reception, the detector yields an attenuated but accurate reproduction of the baseband signal. However, ϕ typically fluctuates unpredictably over time due to channel variability. This leads to a corresponding random variation in the detector's output, which is not desirable.

To keep the local oscillator perfectly synchronized with the original carrier, we need more sophisticated circuitry than what’s shown in Figure 2. We’ll examine one such circuit in the next section.

 

The Costas Loop

One method for achieving phase-coherent demodulation is to use a phase-locked loop. The resulting demodulation circuit, known as the Costas loop, can be seen in Figure 4.

 

Figure 4. The Costas loop.

 

This circuit contains two detector paths:

• The upper path, known as the in-phase detector or I-channel.
• The lower path, known as the quadrature detector or Q-channel.

Like the basic demodulator in Figure 2, each path includes a multiplier and a low-pass filter. The multiplier on the I-channel’s path is driven by a cosine wave:

$$c_I(t)~=~2\cos(\omega_ct~+~\theta_r)$$

Equation 7.

 

where θ_r is the phase of the local oscillator.

The Q-channel’s multiplier is driven by a sine wave:

$$c_Q(t)~=~2\sin(\omega_ct~+~\theta_r)$$

Equation 8.

 

Another multiplier combines the outputs of the in-phase and quadrature paths, producing a feedback signal that keeps the voltage-controlled oscillator (VCO) sine wave synchronized with the original carrier wave.

 

Operation of the Costas Loop

Let’s follow a signal from the input of Figure 4 to its output. We start with a DSB-SC signal given by:

$$r(t)~=~m(t) \cos(\omega_c t ~+~\theta_i)$$

Equation 9.

 

where θ_i is the phase of the input signal.

The signal passes through the I-channel to the output at node C. In addition, the input signal passes through the Q-channel to node D. We now have two different signals:

$$v_c~=~m(t) \cos(\theta_e) \quad \text{and} \quad v_D~=~m(t)\sin(\theta_e)$$

Equation 10.

 

where θ_e = θ_i – θ_r.

We will use these two signals to provide feedback to the VCO. We begin by multiplying the signals at nodes C and D together, producing the following at node E:

$$v_E~=~0.5m^2(t) \sin(2 \theta_e)$$

Equation 11.

 

After that, the signal passes through another low-pass filter, creates the feedback signal at node F:

$$v_F~=~R\sin(2 \theta_e)$$

Equation 12.

 

where R is the DC component of 0.5m²(t). This is applied to the input of the VCO, which has a quiescent frequency of ⍵_c.

The feedback circuit automatically corrects any phase error between the local oscillator and the original carrier. When the phase error is zero (θ_e = 0), the upper arm produces the message signal (m(t)) and the lower path’s output reduces to zero.

 

VCO Phase Error Correction

To understand how the circuit minimizes phase error, let’s assume that the phase of the local oscillator slightly deviates from the ideal value. Assuming that the phase error is small, the signal at node E can be approximated as:

$$v_E ~=~ 0.5m^2(t)\sin(2 \theta_e) ~\approx~ 0.5m^2(t) ~\times~ 2 \theta_e~=~m^2(t) ~\times~ ( \theta_i ~-~ \theta_r )$$

Equation 13.

 

In the equation above, we see that v_E is proportional to the phase error. In other words, the polarity and amplitude of v_E depend on the sign and amplitude of θ_e. By passing v_E through a low-pass filter, we obtain a DC control signal for tuning the VCO.

 

Synchronous Demodulation Using a Pilot Carrier

Another way of addressing phase error is to incorporate a low-level carrier into the transmitted signal. This carrier component, known as a pilot tone, serves as a phase reference for synchronous demodulation at the receiver. Figure 5 shows a double-sideband transmitter that includes a pilot tone in the transmitted signal.

 

Figure 5. A multiplier and adder create a DSB signal with a transmitted carrier.

 

In the figure above, the carrier is scaled by a factor of k and then added to the output signal. The scaling factor allows us to control the power of the pilot tone relative to the information-bearing signal components. The receiver (Figure 6) uses a narrow-band filter to extract the pilot tone, which is then multiplied by the received signal to perform demodulation.

 

Figure 6. A receiver configured to extract the pilot tone for phase-coherent demodulation.

 

Note that this does not qualify as DSB-SC modulation. Because the carrier is present in the modulated signal spectrum, this doesn’t qualify as a suppressed-carrier technique. The drawback of adding a pilot tone is that it allocates a fraction of the transmitted signal's power to the carrier, which doesn’t convey any message information.

 

Wrapping Up

We have seen the importance of synchronizing the local carrier with the received signal. Phase differences can result in significantly attenuated outputs. In the worst-case scenario, a phase discrepancy of 90 degrees can reduce the output to zero.

Circuits like the Costas loop use feedback to minimize this phase error and maximize the received message signal amplitude for DSB-SC signals. Alternatively, we can incorporate a low-level pilot carrier into the transmitted signal to serve as a phase reference for synchronous demodulation at the receiver.

 

This article is Part 9 of a series on amplitude modulation in RF systems. A complete list of articles in this series is provided below:

1. Introduction to Modulation Techniques in RF Systems
2. Understanding Double-Sideband Suppressed-Carrier Modulation
3. Understanding Conventional Amplitude Modulation
4. Understanding the Square-Law Modulator for Generating AM Signals
5. Introduction to the Balanced Modulator for AM Signals
6. How Do Switching Modulators Generate AM Signals?
7. Understanding How Ring Modulators Produce AM Signals
8. Four Interesting AM Modulation Circuits You Should Know About
9. Demodulating Double-Sideband AM Signals
10. Introduction to Single-Sideband Modulation: The Filter Method
11. The Phasing Method and Hilbert Transforms for Single-Sideband Modulation
12. A Visual Approach to Understanding the Phasing Method for SSB Modulation
13. How Phasors Help Us Understand Bandpass Signals
14. Introduction to Weaver’s Method for SSB Signal Generation
15. Exploring the Operation of the Weaver Modulator for Single-Sideband Modulation

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All images used courtesy of Steve Arar