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Electronics β 96032
Alessandro SpinelliPhone: (02 2399) [email protected] home.deib.polimi.it/spinelli
Amplitude Modulation and Synchronous Detection
Alessandro Spinelli β Electronics 96032
Slides are supplementary material and are NOT a
replacement for textbooks and/or lecture notes
Disclaimer 2
Alessandro Spinelli β Electronics 96032
β’ Itβs time to begin a discussion on the techniques for improvingS/N
β’ Noise-reduction techniques obviously depend on the type of signal and of noise:
Purpose of the lesson 3
NoiseHF (White) LF (flicker)
Sign
al LF (constant) done this lessonHF (pulse) done done
Alessandro Spinelli β Electronics 96032
β’ Amplitude modulation and demodulationβ’ The weighting function
Outline 4
Alessandro Spinelli β Electronics 96032
β’ If the signal is at DC level and buried in LF noise, HP or BP filters become useless
β’ If we could move the signal spectrum to higher frequencies, we would improve ππ/ππ
β’ A high-Q BP filter could then be used to recover the signal
The problem 5
Alessandro Spinelli β Electronics 96032
β’ The amplitude of a carrier wave ππ π‘π‘ is modified by a modulatingsignal π₯π₯ π‘π‘
β’ Originally developed for telephone and radio communication
Amplitude modulation (AM) 6
ππ π‘π‘ = π₯π₯ π‘π‘ ππ(π‘π‘)ππ ππ = ππ ππ β πΆπΆ(ππ)
ππ π‘π‘πΆπΆ(ππ)
π₯π₯ π‘π‘ππ(ππ)
Alessandro Spinelli β Electronics 96032
β’ We consider a sinusoidal carrierππ π‘π‘ = π΄π΄cos πππππ‘π‘ + ππππ β
πΆπΆ ππ =π΄π΄2
πππππππππΏπΏ ππ β ππππ + ππβπππππππΏπΏ ππ + ππππβ’ The modulated signal is then
ππ π‘π‘ = π₯π₯ π‘π‘ ππ π‘π‘ β
ππ ππ = ππ ππ β πΆπΆ ππ =π΄π΄2
ππππππππππ ππ β ππππ + ππβππππππππ ππ + ππππ
Time and frequency domains 7
Alessandro Spinelli β Electronics 96032
Waveforms and spectra 8
From [1]
ππ
ππ
ππ
ππππβππππ
ππππβππππ
|ππ ππ |
|ππ ππ |
|πΆπΆ ππ |
Alessandro Spinelli β Electronics 96032
ππ π‘π‘ = ππ π‘π‘ ππ π‘π‘ = π₯π₯ π‘π‘ ππ2 π‘π‘ = π₯π₯ π‘π‘ π΄π΄2cos2 πππππ‘π‘ + ππππ
=π΄π΄2
2π₯π₯ π‘π‘ 1 + cos 2πππππ‘π‘ + 2ππππ
π·π· ππ =π΄π΄2
2ππ ππ +
π΄π΄2
4ππππ2ππππππ ππ β 2ππππ + ππβππ2ππππππ ππ + 2ππππ
Demodulation 9
ππ π‘π‘ = ππ π‘π‘ ππ(π‘π‘)
π·π· ππ = ππ ππ β πΆπΆ(ππ)
ππ π‘π‘πΆπΆ(ππ)
ππ π‘π‘ππ(ππ) LPF π¦π¦ π‘π‘
ππ(ππ)
ππ ππ
Alessandro Spinelli β Electronics 96032
Frequency-domain view 10
ππ
ππ
ππ
2ππππβ2ππππ
ππππβππππ
|ππ ππ |
|ππ ππ |
|π·π· ππ ||ππ ππ |
π΄π΄
π΄π΄/2π΄π΄/2
π΄π΄2/2
π΄π΄2/4π΄π΄2/4
Alessandro Spinelli β Electronics 96032
β’ We consider demodulating withπ€π€π π π‘π‘ = π΅π΅cos (ππππ+Ξππ)π‘π‘ + ππππ + Ξππ
β’ The low-frequency term (small Ξππ) isπ¦π¦ π‘π‘ = π΄π΄π΅π΅π₯π₯ π‘π‘ cos(Ξπππ‘π‘ + Ξππ)/2
Phase error β signal reduction Frequency error β oscillating behavior
β’ The reference must be locked in frequency and phase to the carrier β synchronous detection
β’ The demodulator is also called phase-sensitive detector (PSD)
Frequency and phase errors 11
Alessandro Spinelli β Electronics 96032
β’ Amplitude modulation and demodulationβ’ The weighting function
Outline 12
Alessandro Spinelli β Electronics 96032
π¦π¦ π‘π‘ = οΏ½ππ ππ π€π€πΏπΏπΏπΏ π‘π‘, ππ ππππ = οΏ½π₯π₯ ππ π€π€π π ππ π€π€πΏπΏπΏπΏ π‘π‘, ππ ππππ
π€π€ π‘π‘, ππ = π€π€π π ππ π€π€πΏπΏπΏπΏ π‘π‘, ππππ π‘π‘, ππ = πππ π ππ βπππΏπΏπΏπΏ(π‘π‘, ππ)
PSD weighting function 13
ππ π‘π‘
π·π· ππ
π€π€π π π‘π‘πππ π (ππ)
π₯π₯ π‘π‘ππ(ππ) LPF
π¦π¦ π‘π‘
ππ(ππ)
Alessandro Spinelli β Electronics 96032
Time/frequency domains 14
ππ
ππ
πππ‘π‘
π‘π‘
π‘π‘
π€π€π π ππ
π€π€πΏπΏπΏπΏ π‘π‘, ππ
π€π€ π‘π‘, ππ
ππ
ππ
ππ
|πππ π ππ |
|πππΏπΏπΏπΏ ππ |
|ππ ππ |
Alessandro Spinelli β Electronics 96032
β’ The filter is time-variant even if LPF is LTI β remember that
π¦π¦ π‘π‘ = οΏ½π₯π₯ ππ π€π€ π‘π‘, ππ ππππ = οΏ½ππ ππ ππβ π‘π‘, ππ ππππ
β ππ(ππ) is NOT equal to ππ ππ ππ(π‘π‘, ππ)β’ For the particular case of a PSD with LTI LPF, we have instead
ππ ππ = πππΏπΏπΏπΏ ππ (ππ ππ βπππ π ππ )β’ If π€π€π π π‘π‘ is a periodic signal, |ππ ππ | represents the frequency
components that give contributions in the baseband
Notes 15
Alessandro Spinelli β Electronics 96032
Weghting function interpretation 16
ππ
|ππ ππ |
ππ
|ππ ππ |
ππ
|ππ ππ |
ππ
|ππ ππ |
ππ
|ππ ππ |
ππ
|ππ ππ |
Alessandro Spinelli β Electronics 96032
β’ Letβs take a simple case of constant signal β the input of the PSD is then π₯π₯ π‘π‘ = π΄π΄cos πππππ‘π‘
β’ If LPF is an LTI integratorπ€π€πΏπΏπΏπΏ π‘π‘, ππ = πΎπΎu π‘π‘ β ππ β π€π€ π‘π‘, ππ β π₯π₯ ππ
β PSD is the optimum filter!β’ In the general case, PSD is quasi-optimum, but more flexible Reference frequency set externally BW of LPF can accomodate different signals
PSD as optimum filter 17
Alessandro Spinelli β Electronics 96032
β’ For the case of the LTI integrator we have
π¦π¦ π‘π‘ = πΎπΎοΏ½ββ
π‘π‘π₯π₯ ππ π€π€π π ππ ππππ β πΎπΎπ₯π₯π€π€π π (0)
β’ π¦π¦ π‘π‘ is an estimate of the cross-correlation between input and referencesignals β maximum output is achieved when π₯π₯ π‘π‘ has the same frequencyand phase as π€π€π π π‘π‘
β’ For, say, a simple RC filter we have
π¦π¦ π‘π‘ =1πππΉπΉοΏ½ββ
π‘π‘π₯π₯ ππ π€π€π π ππ ππβ
π‘π‘βπππππΉπΉ ππππ
which is again πΎπΎπ₯π₯π€π€π π (0) estimated over a time πππΉπΉ
Another look at the weighting function 18
Alessandro Spinelli β Electronics 96032
β’ In the frequency domain we have
ππ ππ βπππ π ππ = οΏ½ππ(ππ)πππ π ππ β ππ ππππ
β’ The output LPF selects the components around ππ = 0, i.e.
ππ ππ β οΏ½ππ ππ πππ π βππ ππππ = οΏ½ππ ππ πππ π β ππ ππππ
We find again the correlation behavior!
Frequency domain 19
Alessandro Spinelli β Electronics 96032
The output noise of the PSD is non-stationary (actuallycyclostationary) even if the input noise is
Output noise 20
π π ππππ π‘π‘, ππ
ππππ π‘π‘,ππ
π€π€π π π‘π‘πππ π (ππ)
π π π₯π₯π₯π₯ πππππ₯π₯(ππ) LPF
π π π¦π¦π¦π¦ π‘π‘, ππ
πππ¦π¦(π‘π‘,ππ)
From [2]
Alessandro Spinelli β Electronics 96032
π π ππππ π‘π‘, π‘π‘ + ππ = ππππ π‘π‘ ππππ(π‘π‘ + ππ) = πππ₯π₯ π‘π‘ πππ₯π₯ π‘π‘ + ππ π€π€π π π‘π‘ π€π€π π π‘π‘ + ππ= π π π₯π₯π₯π₯(ππ)π€π€π π π‘π‘ π€π€π π π‘π‘ + ππ
β’ In particular, ππππ2(π‘π‘) = πππ₯π₯2π€π€π π 2(π‘π‘)β’ Since the output filter averages over many periods of π€π€π π , we consider
the time averageπ π ππππ π‘π‘, π‘π‘ + ππ = π π π₯π₯π₯π₯(ππ) π€π€π π π‘π‘ π€π€π π π‘π‘ + ππ
= π π π₯π₯π₯π₯(ππ) limππββ
12ππ
οΏ½βππ
πππ€π€π π π‘π‘ π€π€π π π‘π‘ + ππ πππ‘π‘ = π π π₯π₯π₯π₯ ππ πΎπΎπ€π€π π π€π€π π (ππ)
Output noise autocorrelation 21
Alessandro Spinelli β Electronics 96032
π€π€π π π‘π‘ = π΅π΅ cosπππππ‘π‘
πΎπΎπ€π€π π π€π€π π ππ = limππββ
π΅π΅2
2πποΏ½βππ
ππcosπππππ‘π‘ cosππππ(π‘π‘ + ππ)πππ‘π‘
= limππββ
π΅π΅2
2πποΏ½βππ
ππcosπππππ‘π‘ (cosπππππ‘π‘ cosππππππ β sinπππππ‘π‘ sinππππππ)πππ‘π‘ =
π΅π΅2 cosππππππ limππββ
12ππ
οΏ½βππ
ππcos2 πππππ‘π‘ πππ‘π‘ =
π΅π΅2
2cosππππππ
Reference signal time correlation 22
Alessandro Spinelli β Electronics 96032
β’ In the frequency domain we have (average spectral density!)ππππ ππ = πππ₯π₯ ππ β πππ€π€π π (ππ)
β’ π€π€π π is a power signal β we consider the truncated signal π€π€π π ππ, whose transform is
πππ π ππ ππ =
π΅π΅2 πΏπΏ ππ β ππππ + πΏπΏ(ππ + ππππ) β 2ππsinc 2ππππππ
=π΅π΅2 2ππ sinc 2ππ ππ β ππππ ππ + sinc 2ππ ππ + ππππ ππ
πππ π ππ ππ 2 =
π΅π΅2
4 2ππ 2 sinc2 2ππ ππ β ππππ ππ + sinc2 2ππ ππ + ππππ ππ
πππ€π€π π ππ = limππββ
12ππ πππ π
ππ ππ 2 =π΅π΅2
4 πΏπΏ ππ β ππππ + πΏπΏ(ππ + ππππ)
which is obviously the Fourier transform of π΅π΅2
2cosππππππ
Frequency domain 23
Alessandro Spinelli β Electronics 96032
β’ The reference signal is
π€π€π π π‘π‘ = π΅π΅cos πππππ‘π‘ β πππ π ππ =π΅π΅2πΏπΏ ππ β ππππ + πΏπΏ ππ + ππππ
β’ Its time correlation is
πΎπΎπ€π€π π π€π€π π ππ =π΅π΅2
2cos ππππππ β πππ€π€π π ππ =
π΅π΅2
4πΏπΏ ππ β ππππ + πΏπΏ ππ + ππππ
β’ The output noise becomes
π π ππππ ππ = π π π₯π₯π₯π₯ πππ΅π΅2
2cos ππππππ β ππππ ππ =
π΅π΅2
4πππ₯π₯ ππ β ππππ + πππ₯π₯ ππ + ππππ
Wrap up 24
Alessandro Spinelli β Electronics 96032
Ex: flicker noise spectra 25
ππ
ππππβππππ
ππππ ππ
ππ
πππ₯π₯ ππ
π΅π΅2
4 πππ₯π₯ ππ β πππππ΅π΅2
4 πππ₯π₯ ππ + ππππ
πππΏπΏπΏπΏ π‘π‘, ππ 2
Alessandro Spinelli β Electronics 96032
These are the input noise components that will contribute to the output noise β remember the meaning of the weighting function!
Output noise 26
ππππβππππ
ππ
πππ₯π₯ ππ ππ ππ 2
Alessandro Spinelli β Electronics 96032
β’ We consider the equivalent noise bandwidth of πππΏπΏπΏπΏ π‘π‘, ππ , π΅π΅ππππ
β’ We approximate ππππ ππ with ππππ 0 over π΅π΅ππππ (narrow-band output filter)
β’ The output mean square noise is then
πππ¦π¦2 = οΏ½ππππ ππ πππΏπΏπΏπΏ π‘π‘, ππ 2ππππ β ππππ 0 2π΅π΅ππππ
= 2π΅π΅πππππ΅π΅2
4πππ₯π₯ βππππ + πππ₯π₯ ππππ = π΅π΅2π΅π΅πππππππ₯π₯ ππππ
Output rms noise 27
Alessandro Spinelli β Electronics 96032
β’ We consider a constant (modulated) signalπ₯π₯ π‘π‘ = π΄π΄cos πππππ‘π‘
β’ Output signal and noise are
π¦π¦ π‘π‘ =π΄π΄π΅π΅2
πππ¦π¦2 = π΅π΅2π΅π΅πππππππ₯π₯ ππππππππ
=π΄π΄
4πππ₯π₯ ππππ π΅π΅ππππ
πΊπΊ/π΅π΅ ratio 28
(no phase errors)
Alessandro Spinelli β Electronics 96032
β’ If only LPF were used, ππ/ππ would beππππ
=π΄π΄
2πππ₯π₯ 0 π΅π΅ππππ
β’ Synchronous detection is useful only if πππ₯π₯ 0 β« πππ₯π₯ ππππβ’ The modulation stage must be inserted before the relevant LF
noise sources (usually the amplifiers)
Remarks 29
Alessandro Spinelli β Electronics 96032
Wheatstone bridge with LIA 30
Vo
R
R R(1+x)
R
LIA
ππsin(πππππ‘π‘)
Alessandro Spinelli β Electronics 96032
Low-light measurement with LIA 31
From [3]
Alessandro Spinelli β Electronics 96032
1. http://cnx.org/content/m45974/latest/2. http://home.deib.polimi.it/cova/elet/lezioni/SSN08c_Filters-
BPF3.pdf3. http://en.wikipedia.org/wiki/Lock-in_amplifier
References 32