# Coherent Detection of DSB-SC AM wave

One method to demodulate DSB-SC(Double Sideband-Suppressed Carrier) AM(Amplitude Modulation) wave is to use coherent detection or synchronous detection. In this method, the DSB-SC AM signal is multiplied with locally generated carrier signal at the receiver which is synchronous in frequency and phase with the carrier signal used at the transmitter. Here we will illustrate coherent detection of DSB-SC AM signal using AD633 multiplier IC(Integrated Circuit).

The following shows the circuit diagram of coherent detection of DSB-SC.

In the above circuit we have AM modulator which generates DSB-SC signal using the AD633 analog multiplier IC(Integrated Circuit). The DSB-SC signal is then amplified using BJT amplifier circuit. This amplified DSB-SC AM signal is then transmitted. At the receiver, the received DSB-SC signal is multiplied with the locally generated carrier signal using the local oscillator. The device that performs the multiplication is called product modulator and here AD633 IC is used as the product modulator. The output of the product modulator is low pass filter using 2nd order RC low pass filter.

The following shows the AM modulator circuit.

The message signal is Vtxm which has frequency of 1KHz and amplitude of 1.2V. The carrier signal  is Vc which has frequency of 120KHz and amplitude of 1.2V. These two are applied to X1 and Y1 pins of the AD633 IC. The X2,Y2 and Z pins are grounded. The output appears on W pin which is DSB-SC signal which is described by the following equation.

$$W = \frac{(X1-X2)(Y1-Y2)}{10}+Z$$

since Z=0 we have,

$$W = \frac{(X1-X2)(Y1-Y2)}{10}$$

here $$X1 = A_m cos(w_m t)$$ is the message signal and $$Y1 = A_c cos(w_c t)$$ is the carrier signal.

Thus, $$W = \frac{A_m cos(w_m t) \times A_c cos(w_c t)}{10}$$

or,  $$W = \frac{A_m A_c}{10} cos(w_m t) cos(w_c t)$$  ------------->(1)

is the resulting DSB-SC signal generated at the transmitter.

This DSB-SC signal is then amplified using the BJT amplifier circuit shown below.

Here 2N2222 bipolar junction transistor was used. The amplifier was biased using voltage divider biasing method. The component values can be calculated using the online BJT amplifier design calculator.

At the receiver we have the product modulator IC AD633 which is supplied with the received DSB-SC signal and the local oscillator carrier signal. This circuit along with the 2nd order low pass filter is shown below.

In the above coherent detection circuit diagram, the received DSB-SC signal is applied to X1 pin and the locally generated carrier signal is applied to the Y1 pin. The X2 and Y2 pins are grounded and the summing input pin Z is also grounded. The output of the AD633 integrated circuit based product modulator appears at the pin W which contains the message signal and high frequencies components.

From equation(1) above, the received DSB-SC AM wave is,

$$s(t) = \frac{A_m A_c}{10} cos(w_m t) cos(w_c t)$$

This is multiplied with the locally generated carrier wave,

$$c(t) = cos(w_c t + \phi)$$

where $$\phi$$ is the phase difference between the locally generated carrier signal and the carrier signal used at the transmitter.

The output from the product modulator is,

$$v(t) = s(t) \times c(t)$$

or, $$v(t) = \frac{A_m A_c}{10} cos(w_m t) cos(w_c t) cos(w_c t + \phi)$$

or,  $$v(t) = \frac{A_m A_c}{10} cos(w_m t) cos(w_c t) [cos(w_c t) cos\phi - sin(w_c t) sin\phi]$$

or,  $$v(t) = \frac{A_m A_c}{10} cos(w_m t) cos\phi cos^2(w_c t) - \frac{A_m A_c}{10} cos(w_m t) sin\phi cos(w_c t) sin(w_c t)]$$

or,  $$v(t) = \frac{A_m A_c}{10} cos(w_m t) cos\phi [\frac{1+cos2(w_c t)}{2}] - \frac{A_m A_c}{10} cos(w_m t) sin\phi [\frac{sin2(w_c t)}{2}]$$

or,  $$v(t) = \frac{A_m A_c}{10} cos(w_m t) cos\phi [\frac{1+cos2(w_c t)}{2}] - \frac{A_m A_c}{20} cos(w_m t) sin\phi sin2(w_c t)$$

or,  $$v(t) = (\frac{A_m A_c}{20}cos\phi) cos(w_m t) +\frac{A_m A_c}{20} cos(w_m t) cos\phi cos2(w_c t) - \frac{A_m A_c}{20} cos(w_m t) sin\phi sin2(w_c t)$$

The above equation describes the output from the product modulator. It contains the message signal $$cos(w_m t)$$ multiplied by scaling constant $$(\frac{A_m A_c}{20}cos\phi)$$ and higher frequency components. The higher frequency component is removed by the low pass filter.

The output from the filter is thus,

or,  $$v(t) = (\frac{A_m A_c}{20}cos\phi) cos(w_m t)$$

This shows that the output is proportional to the message signal and that the amplitude of the demodulated signal depends upon the phase $$\phi$$. The amplitude is maximum when $$\phi = 0$$ and amplitude is minimum when when $$\phi = \pm \frac{\pi}{2}$$. When $$\phi = \pm \frac{\pi}{2}$$ the demodulated signal is zero and is called quadrature null effect. This difference in phase causes attenuation of the DSB-SC AM signal which is proportional to $$cos\phi$$. If the phase is constant then we get undistorted version of the original message but due to noise in the communication channel, the phase changes randomly and we get varying attenuated message signal.

In the above circuit implementation, the low pass filter has resistor value of 750Ohm and capacitor of 10nF. Using these values the cutoff frequency is around 22KHz which is upper limit of the audio signals. Also two RC filter is cascaded to get better result.

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