# Instrumentation & Measurement Magazine 24-2 - 73

```Fig. 1. (a) The power spectral density (PSD) of a measured signal; (b) The desired signal (in green) and noise (in red); (c) Adding a narrow band filter to increase
signal-to-noise ratio; (d) Decreasing the band pass to further increase signal-to-noise ratio. The unit on the y-axis is arbitrary.

Synchronous Demodulation Concepts
The basis of this method is that if two sinusoidal signals with
two different frequencies are multiplied, the mean value of the
resulting function will be zero. In other words, the dc component of the resulting signal will be zero; we can extract a weak
signal by multiplying the measured signal by a sinusoidal signal at a specific frequency following by time-averaging. The
response of the system will be zero except in that particular
frequency. To explain how a synchronous demodulator works,
consider the product of two harmonic signals:
A1 cos  2 f1t     A2 cos  2 f2t 

A1 A2 
cos 2  f1  f2     cos 2  f1  f2  t  

2 







(1)



where A1 and A2 are the amplitudes; f1 and f2 are the corresponding frequencies, and ϕ is the phase difference. Depending on
the values of f1, f2, and ϕ, we can consider two different cases.
In the case of f1 = f2, as we can see in (1), the product consists of
a dc component with the amplitude of (A1A2)/2 × cosϕ and an
ac component at the frequency 2f1. Fig. 2a shows the product
of two phase-coherent harmonic functions with identical frequency. In the second case f1 ≠ f2, there is no dc component, and
April 2021

the product of the signals can be written as two harmonics with
two different frequencies (Fig. 2b).
In a synchronous demodulator, a reference (usually sinusoidal) signal is used to excite the DUT whose output is then
measured as a second signal. For a linear DUT, the frequencies
of the input and output will be the same, but the amplitude
and phase may be affected. In the next step, to extract the dc
component of the signal, we pass the product signal through
a low-pass filter for time averaging. Equation (2) shows how
time averaging leads to a final dc signal that is proportional to
the DUT output signal:
T

A1 A2 
cos 2  f1  f2     cos 2  f1  f 2  t   

2 
0
(2)
 A1 A2
cos  f1  f2

 2

f1  f 2
0


1
T  T

lim











As a result, the portion of the spectrum with f1 ≠ f2 is filtered,
and only the frequency component at the reference frequency
f1 = f2 survives as a dc value with an amplitude that is commensurate with the measured signal. Noise frequency components
(e.g., 60 Hz line frequencies) are filtered by synchronous

IEEE Instrumentation & Measurement Magazine	73

```

# Instrumentation & Measurement Magazine 24-2

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