amplitude modulation and synchronous detection

Electronics – 96032

Alessandro SpinelliPhone: (02 2399) [email protected] home.deib.polimi.it/spinelli

Amplitude Modulation and Synchronous Detection

Alessandro Spinelli – Electronics 96032

Slides are supplementary material and are NOT a

replacement for textbooks and/or lecture notes

Disclaimer 2


• It’s time to begin a discussion on the techniques for improvingS/N

• 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


• Amplitude modulation and demodulation• The weighting function

Outline 4


• 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 𝑆𝑆/𝑁𝑁

• A high-Q BP filter could then be used to recover the signal

The problem 5


• The amplitude of a carrier wave 𝑐𝑐 𝑡𝑡 is modified by a modulatingsignal 𝑥𝑥 𝑡𝑡

• Originally developed for telephone and radio communication

Amplitude modulation (AM) 6

𝑚𝑚 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝑐𝑐(𝑡𝑡)𝑀𝑀 𝑓𝑓 = 𝑋𝑋 𝑓𝑓 ∗ 𝐶𝐶(𝑓𝑓)

𝑐𝑐 𝑡𝑡𝐶𝐶(𝑓𝑓)

𝑥𝑥 𝑡𝑡𝑋𝑋(𝑓𝑓)


• We consider a sinusoidal carrier𝑐𝑐 𝑡𝑡 = 𝐴𝐴cos 𝜔𝜔𝑐𝑐𝑡𝑡 + 𝜙𝜙𝑐𝑐 ⇔

𝐶𝐶 𝑓𝑓 =𝐴𝐴2

𝑒𝑒𝑗𝑗𝜙𝜙𝑐𝑐𝛿𝛿 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝑒𝑒−𝑗𝑗𝜙𝜙𝑐𝑐𝛿𝛿 𝑓𝑓 + 𝑓𝑓𝑐𝑐• The modulated signal is then

𝑚𝑚 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝑐𝑐 𝑡𝑡 ⇔

𝑀𝑀 𝑓𝑓 = 𝑋𝑋 𝑓𝑓 ∗ 𝐶𝐶 𝑓𝑓 =𝐴𝐴2

𝑒𝑒𝑗𝑗𝜙𝜙𝑐𝑐𝑋𝑋 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝑒𝑒−𝑗𝑗𝜙𝜙𝑐𝑐𝑋𝑋 𝑓𝑓 + 𝑓𝑓𝑐𝑐

Time and frequency domains 7


𝑑𝑑 𝑡𝑡 = 𝑚𝑚 𝑡𝑡 𝑐𝑐 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝑐𝑐2 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝐴𝐴2cos2 𝜔𝜔𝑐𝑐𝑡𝑡 + 𝜙𝜙𝑐𝑐

=𝐴𝐴2

2𝑥𝑥 𝑡𝑡 1 + cos 2𝜔𝜔𝑐𝑐𝑡𝑡 + 2𝜙𝜙𝑐𝑐

𝐷𝐷 𝑓𝑓 =𝐴𝐴2

2𝑋𝑋 𝑓𝑓 +

𝐴𝐴2

4𝑒𝑒𝑗𝑗2𝜙𝜙𝑐𝑐𝑋𝑋 𝑓𝑓 − 2𝑓𝑓𝑐𝑐 + 𝑒𝑒−𝑗𝑗2𝜙𝜙𝑐𝑐𝑋𝑋 𝑓𝑓 + 2𝑓𝑓𝑐𝑐

Demodulation 9

𝑑𝑑 𝑡𝑡 = 𝑚𝑚 𝑡𝑡 𝑐𝑐(𝑡𝑡)

𝐷𝐷 𝑓𝑓 = 𝑀𝑀 𝑓𝑓 ∗ 𝐶𝐶(𝑓𝑓)

𝑐𝑐 𝑡𝑡𝐶𝐶(𝑓𝑓)

𝑚𝑚 𝑡𝑡𝑀𝑀(𝑓𝑓) LPF 𝑦𝑦 𝑡𝑡

𝑌𝑌(𝑓𝑓)

𝑌𝑌 𝑓𝑓


Frequency-domain view 10

𝑓𝑓

𝑓𝑓

𝑓𝑓

2𝑓𝑓𝑐𝑐−2𝑓𝑓𝑐𝑐



|𝑀𝑀 𝑓𝑓 |

|𝐷𝐷 𝑓𝑓 ||𝑌𝑌 𝑓𝑓 |

𝐴𝐴

𝐴𝐴/2𝐴𝐴/2

𝐴𝐴2/2

𝐴𝐴2/4𝐴𝐴2/4


• We consider demodulating with𝑤𝑤𝑅𝑅 𝑡𝑡 = 𝐵𝐵cos (𝜔𝜔𝑐𝑐+Δ𝜔𝜔)𝑡𝑡 + 𝜙𝜙𝑐𝑐 + Δ𝜙𝜙

• The low-frequency term (small Δ𝜔𝜔) is𝑦𝑦 𝑡𝑡 = 𝐴𝐴𝐵𝐵𝑥𝑥 𝑡𝑡 cos(Δ𝜔𝜔𝑡𝑡 + Δ𝜙𝜙)/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)

Frequency and phase errors 11


• Amplitude modulation and demodulation• The weighting function

Outline 12


𝑦𝑦 𝑡𝑡 = �𝑑𝑑 𝜏𝜏 𝑤𝑤𝐿𝐿𝐿𝐿 𝑡𝑡, 𝜏𝜏 𝑑𝑑𝜏𝜏 = �𝑥𝑥 𝜏𝜏 𝑤𝑤𝑅𝑅 𝜏𝜏 𝑤𝑤𝐿𝐿𝐿𝐿 𝑡𝑡, 𝜏𝜏 𝑑𝑑𝜏𝜏

𝑤𝑤 𝑡𝑡, 𝜏𝜏 = 𝑤𝑤𝑅𝑅 𝜏𝜏 𝑤𝑤𝐿𝐿𝐿𝐿 𝑡𝑡, 𝜏𝜏𝑊𝑊 𝑡𝑡, 𝑓𝑓 = 𝑊𝑊𝑅𝑅 𝑓𝑓 ∗𝑊𝑊𝐿𝐿𝐿𝐿(𝑡𝑡, 𝑓𝑓)

PSD weighting function 13

𝑑𝑑 𝑡𝑡

𝐷𝐷 𝑓𝑓

𝑤𝑤𝑅𝑅 𝑡𝑡𝑊𝑊𝑅𝑅(𝑓𝑓)

𝑥𝑥 𝑡𝑡𝑋𝑋(𝑓𝑓) LPF

𝑦𝑦 𝑡𝑡

𝑌𝑌(𝑓𝑓)


• The filter is time-variant even if LPF is LTI ⇒ remember that

𝑦𝑦 𝑡𝑡 = �𝑥𝑥 𝜏𝜏 𝑤𝑤 𝑡𝑡, 𝜏𝜏 𝑑𝑑𝜏𝜏 = �𝑋𝑋 𝑓𝑓 𝑊𝑊∗ 𝑡𝑡, 𝑓𝑓 𝑑𝑑𝑓𝑓

⇒ 𝑌𝑌(𝑓𝑓) is NOT equal to 𝑋𝑋 𝑓𝑓 𝑊𝑊(𝑡𝑡, 𝑓𝑓)• For the particular case of a PSD with LTI LPF, we have instead

𝑌𝑌 𝑓𝑓 = 𝑊𝑊𝐿𝐿𝐿𝐿 𝑓𝑓 (𝑋𝑋 𝑓𝑓 ∗𝑊𝑊𝑅𝑅 𝑓𝑓 )• If 𝑤𝑤𝑅𝑅 𝑡𝑡 is a periodic signal, |𝑊𝑊 𝑓𝑓 | represents the frequency

components that give contributions in the baseband

Notes 15


Weghting function interpretation 16

𝑓𝑓


𝑓𝑓


𝑓𝑓

|𝑌𝑌 𝑓𝑓 |

𝑓𝑓


𝑓𝑓


𝑓𝑓

|𝑌𝑌 𝑓𝑓 |


• Let’s take a simple case of constant signal ⇒ the input of the PSD is then 𝑥𝑥 𝑡𝑡 = 𝐴𝐴cos 𝜔𝜔𝑐𝑐𝑡𝑡

• If LPF is an LTI integrator𝑤𝑤𝐿𝐿𝐿𝐿 𝑡𝑡, 𝜏𝜏 = 𝐾𝐾u 𝑡𝑡 − 𝜏𝜏 ⇒ 𝑤𝑤 𝑡𝑡, 𝜏𝜏 ∝ 𝑥𝑥 𝜏𝜏

⇒ PSD is the optimum filter!• 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


• For the case of the LTI integrator we have

𝑦𝑦 𝑡𝑡 = 𝐾𝐾�−∞

𝑡𝑡𝑥𝑥 𝜏𝜏 𝑤𝑤𝑅𝑅 𝜏𝜏 𝑑𝑑𝜏𝜏 ≈ 𝐾𝐾𝑥𝑥𝑤𝑤𝑅𝑅(0)

• 𝑦𝑦 𝑡𝑡 is an estimate of the cross-correlation between input and referencesignals ⇒ maximum output is achieved when 𝑥𝑥 𝑡𝑡 has the same frequencyand phase as 𝑤𝑤𝑅𝑅 𝑡𝑡

• For, say, a simple RC filter we have

𝑦𝑦 𝑡𝑡 =1𝑇𝑇𝐹𝐹�−∞

𝑡𝑡𝑥𝑥 𝜏𝜏 𝑤𝑤𝑅𝑅 𝜏𝜏 𝑒𝑒−

𝑡𝑡−𝜏𝜏𝑇𝑇𝐹𝐹 𝑑𝑑𝜏𝜏

which is again 𝐾𝐾𝑥𝑥𝑤𝑤𝑅𝑅(0) estimated over a time 𝑇𝑇𝐹𝐹

Another look at the weighting function 18


• In the frequency domain we have

𝑋𝑋 𝑓𝑓 ∗𝑊𝑊𝑅𝑅 𝑓𝑓 = �𝑋𝑋(𝜈𝜈)𝑊𝑊𝑅𝑅 𝑓𝑓 − 𝜈𝜈 𝑑𝑑𝜈𝜈

• The output LPF selects the components around 𝑓𝑓 = 0, i.e.

𝑌𝑌 𝑓𝑓 ≈ �𝑋𝑋 𝜈𝜈 𝑊𝑊𝑅𝑅 −𝜈𝜈 𝑑𝑑𝜈𝜈 = �𝑋𝑋 𝜈𝜈 𝑊𝑊𝑅𝑅∗ 𝜈𝜈 𝑑𝑑𝜈𝜈

We find again the correlation behavior!

Frequency domain 19


The output noise of the PSD is non-stationary (actuallycyclostationary) even if the input noise is

Output noise 20

𝑅𝑅𝑑𝑑𝑑𝑑 𝑡𝑡, 𝜏𝜏

𝑆𝑆𝑑𝑑 𝑡𝑡,𝑓𝑓

𝑤𝑤𝑅𝑅 𝑡𝑡𝑊𝑊𝑅𝑅(𝑓𝑓)

𝑅𝑅𝑥𝑥𝑥𝑥 𝜏𝜏𝑆𝑆𝑥𝑥(𝑓𝑓) LPF

𝑅𝑅𝑦𝑦𝑦𝑦 𝑡𝑡, 𝜏𝜏

𝑆𝑆𝑦𝑦(𝑡𝑡,𝑓𝑓)

From [2]


𝑅𝑅𝑑𝑑𝑑𝑑 𝑡𝑡, 𝑡𝑡 + 𝜏𝜏 = 𝑛𝑛𝑑𝑑 𝑡𝑡 𝑛𝑛𝑑𝑑(𝑡𝑡 + 𝜏𝜏) = 𝑛𝑛𝑥𝑥 𝑡𝑡 𝑛𝑛𝑥𝑥 𝑡𝑡 + 𝜏𝜏 𝑤𝑤𝑅𝑅 𝑡𝑡 𝑤𝑤𝑅𝑅 𝑡𝑡 + 𝜏𝜏= 𝑅𝑅𝑥𝑥𝑥𝑥(𝜏𝜏)𝑤𝑤𝑅𝑅 𝑡𝑡 𝑤𝑤𝑅𝑅 𝑡𝑡 + 𝜏𝜏

• In particular, 𝑛𝑛𝑑𝑑2(𝑡𝑡) = 𝑛𝑛𝑥𝑥2𝑤𝑤𝑅𝑅2(𝑡𝑡)• Since the output filter averages over many periods of 𝑤𝑤𝑅𝑅, we consider

the time average𝑅𝑅𝑑𝑑𝑑𝑑 𝑡𝑡, 𝑡𝑡 + 𝜏𝜏 = 𝑅𝑅𝑥𝑥𝑥𝑥(𝜏𝜏) 𝑤𝑤𝑅𝑅 𝑡𝑡 𝑤𝑤𝑅𝑅 𝑡𝑡 + 𝜏𝜏

= 𝑅𝑅𝑥𝑥𝑥𝑥(𝜏𝜏) lim𝑇𝑇→∞

12𝑇𝑇

�−𝑇𝑇

𝑇𝑇𝑤𝑤𝑅𝑅 𝑡𝑡 𝑤𝑤𝑅𝑅 𝑡𝑡 + 𝜏𝜏 𝑑𝑑𝑡𝑡 = 𝑅𝑅𝑥𝑥𝑥𝑥 𝜏𝜏 𝐾𝐾𝑤𝑤𝑅𝑅𝑤𝑤𝑅𝑅(𝜏𝜏)

Output noise autocorrelation 21


𝑤𝑤𝑅𝑅 𝑡𝑡 = 𝐵𝐵 cos𝜔𝜔𝑐𝑐𝑡𝑡

𝐾𝐾𝑤𝑤𝑅𝑅𝑤𝑤𝑅𝑅 𝜏𝜏 = lim𝑇𝑇→∞

𝐵𝐵2

2𝑇𝑇�−𝑇𝑇

𝑇𝑇cos𝜔𝜔𝑐𝑐𝑡𝑡 cos𝜔𝜔𝑐𝑐(𝑡𝑡 + 𝜏𝜏)𝑑𝑑𝑡𝑡

= lim𝑇𝑇→∞

𝐵𝐵2

2𝑇𝑇�−𝑇𝑇

𝑇𝑇cos𝜔𝜔𝑐𝑐𝑡𝑡 (cos𝜔𝜔𝑐𝑐𝑡𝑡 cos𝜔𝜔𝑐𝑐𝜏𝜏 − sin𝜔𝜔𝑐𝑐𝑡𝑡 sin𝜔𝜔𝑐𝑐𝜏𝜏)𝑑𝑑𝑡𝑡 =

𝐵𝐵2 cos𝜔𝜔𝑐𝑐𝜏𝜏 lim𝑇𝑇→∞

12𝑇𝑇

�−𝑇𝑇

𝑇𝑇cos2 𝜔𝜔𝑐𝑐𝑡𝑡 𝑑𝑑𝑡𝑡 =

𝐵𝐵2

2cos𝜔𝜔𝑐𝑐𝜏𝜏

Reference signal time correlation 22


• In the frequency domain we have (average spectral density!)𝑆𝑆𝑑𝑑 𝑓𝑓 = 𝑆𝑆𝑥𝑥 𝑓𝑓 ∗ 𝑆𝑆𝑤𝑤𝑅𝑅(𝑓𝑓)

• 𝑤𝑤𝑅𝑅 is a power signal ⇒ we consider the truncated signal 𝑤𝑤𝑅𝑅𝑇𝑇, whose transform is

𝑊𝑊𝑅𝑅𝑇𝑇 𝑓𝑓 =

𝐵𝐵2 𝛿𝛿 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝛿𝛿(𝑓𝑓 + 𝑓𝑓𝑐𝑐) ∗ 2𝑇𝑇sinc 2𝜋𝜋𝑓𝑓𝑇𝑇

=𝐵𝐵2 2𝑇𝑇 sinc 2𝜋𝜋 𝑓𝑓 − 𝑓𝑓𝑐𝑐 𝑇𝑇 + sinc 2𝜋𝜋 𝑓𝑓 + 𝑓𝑓𝑐𝑐 𝑇𝑇

𝑊𝑊𝑅𝑅𝑇𝑇 𝑓𝑓 2 =

𝐵𝐵2

4 2𝑇𝑇 2 sinc2 2𝜋𝜋 𝑓𝑓 − 𝑓𝑓𝑐𝑐 𝑇𝑇 + sinc2 2𝜋𝜋 𝑓𝑓 + 𝑓𝑓𝑐𝑐 𝑇𝑇

𝑆𝑆𝑤𝑤𝑅𝑅 𝑓𝑓 = lim𝑇𝑇→∞

12𝑇𝑇 𝑊𝑊𝑅𝑅

𝑇𝑇 𝑓𝑓 2 =𝐵𝐵2

4 𝛿𝛿 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝛿𝛿(𝑓𝑓 + 𝑓𝑓𝑐𝑐)

which is obviously the Fourier transform of 𝐵𝐵2

2cos𝜔𝜔𝑐𝑐𝜏𝜏

Frequency domain 23


• The reference signal is

𝑤𝑤𝑅𝑅 𝑡𝑡 = 𝐵𝐵cos 𝜔𝜔𝑐𝑐𝑡𝑡 ⇔ 𝑊𝑊𝑅𝑅 𝑓𝑓 =𝐵𝐵2𝛿𝛿 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝛿𝛿 𝑓𝑓 + 𝑓𝑓𝑐𝑐

• Its time correlation is

𝐾𝐾𝑤𝑤𝑅𝑅𝑤𝑤𝑅𝑅 𝜏𝜏 =𝐵𝐵2

2cos 𝜔𝜔𝑐𝑐𝜏𝜏 ⇔ 𝑆𝑆𝑤𝑤𝑅𝑅 𝑓𝑓 =

𝐵𝐵2

4𝛿𝛿 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝛿𝛿 𝑓𝑓 + 𝑓𝑓𝑐𝑐

• The output noise becomes

𝑅𝑅𝑑𝑑𝑑𝑑 𝜏𝜏 = 𝑅𝑅𝑥𝑥𝑥𝑥 𝜏𝜏𝐵𝐵2

2cos 𝜔𝜔𝑐𝑐𝜏𝜏 ⇔ 𝑆𝑆𝑑𝑑 𝑓𝑓 =

𝐵𝐵2

4𝑆𝑆𝑥𝑥 𝑓𝑓 − 𝑓𝑓𝑐𝑐 + 𝑆𝑆𝑥𝑥 𝑓𝑓 + 𝑓𝑓𝑐𝑐

Wrap up 24


Ex: flicker noise spectra 25

𝑓𝑓


𝑆𝑆𝑑𝑑 𝑓𝑓

𝑓𝑓

𝑆𝑆𝑥𝑥 𝑓𝑓

𝐵𝐵2

4 𝑆𝑆𝑥𝑥 𝑓𝑓 − 𝑓𝑓𝑐𝑐𝐵𝐵2

4 𝑆𝑆𝑥𝑥 𝑓𝑓 + 𝑓𝑓𝑐𝑐

𝑊𝑊𝐿𝐿𝐿𝐿 𝑡𝑡, 𝑓𝑓 2


These are the input noise components that will contribute to the output noise ⇒ remember the meaning of the weighting function!

Output noise 26


𝑓𝑓

𝑆𝑆𝑥𝑥 𝑓𝑓 𝑊𝑊 𝑓𝑓 2


• We consider the equivalent noise bandwidth of 𝑊𝑊𝐿𝐿𝐿𝐿 𝑡𝑡, 𝑓𝑓 , 𝐵𝐵𝑊𝑊𝑛𝑛

• We approximate 𝑆𝑆𝑑𝑑 𝑓𝑓 with 𝑆𝑆𝑑𝑑 0 over 𝐵𝐵𝑊𝑊𝑛𝑛 (narrow-band output filter)

• The output mean square noise is then

𝑛𝑛𝑦𝑦2 = �𝑆𝑆𝑑𝑑 𝑓𝑓 𝑊𝑊𝐿𝐿𝐿𝐿 𝑡𝑡, 𝑓𝑓 2𝑑𝑑𝑓𝑓 ≈ 𝑆𝑆𝑑𝑑 0 2𝐵𝐵𝑊𝑊𝑛𝑛

= 2𝐵𝐵𝑊𝑊𝑛𝑛𝐵𝐵2

4𝑆𝑆𝑥𝑥 −𝑓𝑓𝑐𝑐 + 𝑆𝑆𝑥𝑥 𝑓𝑓𝑐𝑐 = 𝐵𝐵2𝐵𝐵𝑊𝑊𝑛𝑛𝑆𝑆𝑥𝑥 𝑓𝑓𝑐𝑐

Output rms noise 27


• We consider a constant (modulated) signal𝑥𝑥 𝑡𝑡 = 𝐴𝐴cos 𝜔𝜔𝑐𝑐𝑡𝑡

• Output signal and noise are

𝑦𝑦 𝑡𝑡 =𝐴𝐴𝐵𝐵2

𝑛𝑛𝑦𝑦2 = 𝐵𝐵2𝐵𝐵𝑊𝑊𝑛𝑛𝑆𝑆𝑥𝑥 𝑓𝑓𝑐𝑐𝑆𝑆𝑁𝑁

=𝐴𝐴

4𝑆𝑆𝑥𝑥 𝑓𝑓𝑐𝑐 𝐵𝐵𝑊𝑊𝑛𝑛

𝑺𝑺/𝑵𝑵 ratio 28

(no phase errors)


• If only LPF were used, 𝑆𝑆/𝑁𝑁 would be𝑆𝑆𝑁𝑁

=𝐴𝐴

2𝑆𝑆𝑥𝑥 0 𝐵𝐵𝑊𝑊𝑛𝑛

• Synchronous detection is useful only if 𝑆𝑆𝑥𝑥 0 ≫ 𝑆𝑆𝑥𝑥 𝑓𝑓𝑐𝑐• The modulation stage must be inserted before the relevant LF

noise sources (usually the amplifiers)

Remarks 29


Wheatstone bridge with LIA 30

Vo

R

R R(1+x)

R

LIA

𝑉𝑉sin(𝜔𝜔𝑐𝑐𝑡𝑡)


Low-light measurement with LIA 31

From [3]


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

amplitude modulation and synchronous detection

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