Amplitude modulation and
synchronous detection
Alessandro Spinelli
Tel. (02 2399) 4001
[email protected]
home.deib.polimi.it/spinelli
Slides are supplementary
material and are NOT a
replacement for textbooks
and/or lecture notes
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Alessandro Spinelli
The problem
• If the signal is at DC level and buried in LF
noise, HP or BP filters become useless
• If we could «move» the signal spectrum to
higher frequencies, we would improve S/N
• A high-Q BP filter could then be used to
recover the signal
Elettronica 75513
Alessandro Spinelli
Amplitude modulation (AM)
𝑥 𝑡
𝑋(𝑓)
𝑚 𝑡 = 𝑥 𝑡 𝑐(𝑡)
𝑀 𝑓 = 𝑋 𝑓 ∗ 𝐶(𝑓)
𝑐 𝑡
𝐶(𝑓)
• The amplitude of a carrier wave 𝑐 𝑡 is
modified by a modulating signal 𝑥 𝑡
• Originally developed for telephone and radio
communication
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Alessandro Spinelli
Time and frequency domains
From [1]
|𝑋 𝑓 |
𝑓
|𝐶 𝑓 |
𝑓
−𝑓𝑐
|𝑀 𝑓 |
𝑓𝑐
𝑓
−𝑓𝑐
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Alessandro Spinelli
𝑓𝑐
Time and frequency domains
• We consider a sinusoidal carrier
𝑐 𝑡 = 𝐴cos 𝜔𝑐 𝑡 + 𝜙𝑐 ⇔
𝐴 𝑗𝜙
𝐶 𝑓 =
𝑒 𝑐 𝛿 𝑓 − 𝑓𝑐 + 𝑒 −𝑗𝜙𝑐 𝛿 𝑓 + 𝑓𝑐
2
• The modulated signal is then
𝑚 𝑡 =𝑥 𝑡 𝑐 𝑡 ⇔
𝐴 𝑗𝜙
𝑀 𝑓 =
𝑒 𝑐 𝑋 𝑓 − 𝑓𝑐 + 𝑒 −𝑗𝜙𝑐 𝑋 𝑓 + 𝑓𝑐
2
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Alessandro Spinelli
Demodulation
𝑚 𝑡
𝑀(𝑓)
𝑑 𝑡 = 𝑚 𝑡 𝑐(𝑡)
𝐷 𝑓 = 𝑀 𝑓 ∗ 𝐶(𝑓)
LPF
𝑦 𝑡
𝑌(𝑓)
𝑐 𝑡
𝐶(𝑓)
𝑑 𝑡 = 𝑚 𝑡 𝑐 𝑡 = 𝑥 𝑡 𝑐 2 𝑡 = 𝑥 𝑡 𝐴2 cos 2 𝜔𝑐 𝑡 + 𝜙𝑐
1 + cos(2𝜔𝑐 𝑡 + 2𝜙𝑐 )
2
=𝐴 𝑥 𝑡
2
𝐴2
𝐴2 𝑗2𝜙
𝐷 𝑓 =
𝑋 𝑓 +
𝑒 𝑐 𝑋 𝑓 − 2𝑓𝑐 + 𝑒 −𝑗2𝜙𝑐 𝑋 𝑓 + 2𝑓𝑐
2
4
𝑌 𝑓
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Alessandro Spinelli
Frequency domain view
|𝑋 𝑓 |
𝑓
|𝑀 𝑓 |
𝑓
𝑓𝑐
−𝑓𝑐
|𝐷 𝑓 |
|𝑌 𝑓 |
𝑓
2𝑓𝑐
−2𝑓𝑐
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Alessandro Spinelli
Frequency and phase errors
• We consider demodulating with
𝑐′ 𝑡 = 𝐵cos (𝜔𝑐 +Δ𝜔)𝑡 + 𝜙𝑐 + Δ𝜙
• The low-frequency term (small Δ𝜔) is
𝑦 𝑡 = 𝐴𝐵𝑥 𝑡 cos(Δ𝜔𝑡 + Δ𝜙)/2
– Phase error  signal reduction
– Frequency error  oscillating behavior
• The reference must be locked in frequency and
phase to the carrier  synchronous detection
• The demodulator is also called phase-sensitive
detector (PSD)
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Alessandro Spinelli
PSD weighting function
𝑑 𝑡
𝑥 𝑡
𝑋(𝑓)
𝐷 𝑓
LPF
𝑦 𝑡
𝑌(𝑓)
𝑤𝑅 𝑡
𝑊𝑅 (𝑓)
𝑦 𝑡 =
𝑑 𝜏 𝑤𝐿𝑃 𝑡, 𝜏 𝑑𝜏 =
𝑥 𝜏 𝑤𝑅 𝜏 𝑤𝐿𝑃 𝑡, 𝜏 𝑑𝜏
𝑤 𝑡, 𝜏 = 𝑤𝑅 𝜏 𝑤𝐿𝑃 𝑡, 𝜏
𝑦 𝑡 =
𝑥 𝜏 𝑤 𝑡, 𝜏 𝑑𝜏 =
𝑋 𝑓 𝑊 ∗ 𝑡, 𝑓 𝑑𝑓
𝑊 𝑡, 𝑓 = 𝑊𝑅 𝑓 ∗ 𝑊𝐿𝑃 (𝑡, 𝑓)
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Alessandro Spinelli
Time/frequency domains
|𝑊𝑅 𝑓 |
𝑤𝑅 𝜏
𝑡
𝜏
𝑓
|𝑊𝐿𝑃 𝑓 |
𝑤𝐿𝑃 𝑡, 𝜏
𝑡
𝜏
𝑓
|𝑊 𝑓 |
𝑤 𝑡, 𝜏
𝑡
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𝜏
Alessandro Spinelli
𝑓
Notes
• The filter is time-variant even if LPF is LTI 
remember that 𝑌(𝑓) is NOT 𝑋 𝑓 𝑊(𝑡, 𝑓)
• For the particular case of the PSD is instead
𝑌 𝑓 = 𝑊𝐿𝑃 𝑡, 𝑓 (𝑋 𝑓 ∗ 𝑊𝑅 𝑓 )
• If 𝑤𝑅 𝑡 is a periodic signal, |𝑊 𝑓 |
represents the frequency components that
give contributions in the baseband
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Alessandro Spinelli
PSD as optimum filter
• Let’s take a simple case of constant signal 
the input of the PSD is then 𝑥 𝑡 = 𝐴cos 𝜔𝑐 𝑡
• If LPF is an LTI integrator
𝑤𝐿𝑃 𝑡, 𝜏 = 𝐾u 𝑡 − 𝜏 ⇒ 𝑤 𝑡, 𝜏 ∝ 𝑥 𝜏
• PSD is the optimum filter!
• In the general case PSD is quasi-optimum, but
more flexible
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Alessandro Spinelli
A look at the weighting function
• For the case of the LTI integrator we have
𝑡
𝑦 𝑡 =𝐾
0
𝑥 𝜏 𝑤𝑅 𝜏 𝑑𝜏 ≈ 𝐾𝑥𝑤𝑅 (0)
• 𝑦 𝑡 is an estimate of the cross-correlation between
input and reference signals  maximum output is
achieved when 𝑥 𝑡 has the same frequency and phase
as 𝑤𝑅 𝑡
• For, say, a simple RC filter we have
1 𝑡
𝑦 𝑡 =
𝑥 𝜏 𝑤𝑅 𝜏 𝑒 −(𝑡−𝜏)/𝑇𝐹 𝑑𝜏
𝑇𝐹 0
which is again 𝐾𝑥𝑤𝑅 (0) estimated over a time 𝑇𝐹
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Alessandro Spinelli
Frequency domain
• In the frequency domain we have
𝑋 𝑓 ∗ 𝑊𝑅 𝑓 =
𝑋(𝜈)𝑊𝑅 𝑓 − 𝜈 𝑑𝜈
• The output LPF selects the components around
𝑓 = 0, i.e.
𝑌 𝑓 ≈
𝑋(𝜈)𝑊𝑅 −𝜈 𝑑𝜈
• We find again the correlation behavior!
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Output noise
𝑅𝑥𝑥 𝜏
𝑆𝑥 (𝑓)
𝑅𝑑𝑑 𝑡, 𝜏
𝑆𝑑 𝑡, 𝑓
𝑤𝑅 𝑡
𝑊𝑅 (𝑓)
LPF
𝑅𝑦𝑦 𝑡, 𝜏
𝑌(𝑡, 𝑓)
From [2]
The output noise of the PSD is non-stationary
(actually cyclostationary) even if the input noise is
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Alessandro Spinelli
Output noise autocorrelation
𝑅𝑑𝑑 𝑡, 𝑡 + 𝜏 = 𝑛𝑑 𝑡 𝑛𝑑 (𝑡 + 𝜏)
= 𝑛𝑥 𝑡 𝑛𝑥 𝑡 + 𝜏 𝑤𝑅 𝑡 𝑤𝑅 𝑡 + 𝜏
= 𝑅𝑥𝑥 (𝜏)𝑤𝑅 𝑡 𝑤𝑅 𝑡 + 𝜏
• In particular 𝑛𝑑2 (𝑡) = 𝑛𝑥2 𝑤𝑅2 (𝑡)
• Since the output filter averages over many periods of
𝑤𝑅 , we consider the time average
𝑅𝑑𝑑 𝑡, 𝑡 + 𝜏 = 𝑅𝑥𝑥 (𝜏) 𝑤𝑅 𝑡 𝑤𝑅 𝑡 + 𝜏
1 𝑇
= 𝑅𝑥𝑥 (𝜏) lim
𝑤𝑅 𝑡 𝑤𝑅 𝑡 + 𝜏 𝑑𝑡
𝑇→∞ 2𝑇 −𝑇
= 𝑅𝑥𝑥 𝜏 𝐾𝑤𝑅 𝑤𝑅 (𝜏)
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Alessandro Spinelli
Frequency domain
• In the frequency domain we have then
𝑆𝑑 𝑓 = 𝑆𝑥 𝑓 ∗ 𝑆𝑤𝑅 (𝑓) (Average spectral density!)
• The reference autocorrelation is
𝐵
𝑤𝑅 𝑡 = 𝐵cos 𝜔𝑐 𝑡 ⇔ 𝑊𝑅 𝑓 =
𝛿 𝑓 − 𝑓𝑐 + 𝛿 𝑓 + 𝑓𝑐
2
𝐵2
𝐾𝑤𝑅 𝑤𝑅 𝜏 =
cos 𝜔𝑐 𝜏 ⇔ 𝑆𝑤𝑅 𝑓
2
𝐵2
=
𝛿 𝑓 − 𝑓𝑐 + 𝛿 𝑓 + 𝑓𝑐
4
• The output spectrum becomes
𝐵2
𝑆𝑑 𝑓 =
𝑆𝑥 𝑓 − 𝑓𝑐 + 𝑆𝑥 𝑓 + 𝑓𝑐
4
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Alessandro Spinelli
Ex: flicker noise spectra
𝑆𝑥 𝑓
𝑓
𝐵2
𝑆 𝑓 + 𝑓𝑐
4 𝑥
𝐵2
𝑆 𝑓 − 𝑓𝑐
4 𝑥
𝑆𝑑 𝑓
|𝑊𝐿𝑃 𝑡, 𝑓 |
𝑓
𝑓𝑐
−𝑓𝑐
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Alessandro Spinelli
Output rms noise
• We consider the equivalent noise bandwidth of
𝑊𝐿𝑃 𝑡, 𝑓 , 𝐵𝑊𝑛
• We approximate 𝑆𝑑 𝑓 with 𝑆𝑑 0 over 𝐵𝑊𝑛
(narrow-band output filter)
• The output rms noise is then
𝑛𝑦2 =
𝑆𝑑 𝑓 |𝑊𝐿𝑃 𝑡, 𝑓 |𝑑𝑓 ≈ 𝑆𝑑 0 2𝐵𝑊𝑛
𝐵2
= 2𝐵𝑊𝑛
𝑆𝑥 −𝑓𝑐 + 𝑆𝑥 𝑓𝑐
4
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Alessandro Spinelli
= 𝐵2 𝐵𝑊𝑛 𝑆𝑥 𝑓𝑐
S/N ratio
• We consider a constant signal
𝑥 𝑡 = 𝐴cos 𝜔𝑐 𝑡
• Output signal and noise are
𝐴𝐵
(no phase errors)
𝑦 𝑡 =
2
𝑛𝑦2 = 𝐵2 𝐵𝑊𝑛 𝑆𝑥 𝑓𝑐
𝑆
𝐴
=
𝑁
4𝑆𝑥 𝑓𝑐 𝐵𝑊𝑛
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Alessandro Spinelli
Remarks
• If only LPF is used, S/N would be
𝑆
𝐴
=
𝑁
2𝑆𝑥 0 𝐵𝑊𝑛
• Synchronous detection is useful only if
𝑆𝑥 0 ≫ 𝑆𝑥 𝑓𝑐
• The modulation stage must be inserted before
the relevant LF noise sources (usually the
amplifiers)
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Wheatstone bridge with LIA
R
R
LIA
𝑉sin(𝜔𝑐 𝑡)
R(1+x)
R
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Alessandro Spinelli
Vo
Low-light measurement with LIA
From [3]
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References
1. http://cnx.org/content/m45974/latest/
2. http://home.deib.polimi.it/cova/elet/lezioni/
SSN08c_Filters-BPF3.pdf
3. http://en.wikipedia.org/wiki/Lockin_amplifier
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