Lecture 8 Sample Sample Chapter 8 and 10 Statistic Shot Noise - - PowerPoint PPT Presentation

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Lecture 8

Chapter 8 and 10

ECE243b Lightwave Communications - Spring 2019 Lecture 8 1

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Direct to Baseband Demodulation

Balanced photodetector shown in Figure ?? implements homodyne demodulation

• f only one component of the baseband signal

A separate different balanced photodetector based on a 90-degree hybrid coupler must be used to homodyne demodulate a lightwave waveform directly into a complex-baseband electrical waveform Recall that the real part of the complex-baseband signal sI(t) is the cosine-modulated component of the waveform s(t) The imaginary part sQ(t) is the sine-modulated component of the waveform s(t)

ECE243b Lightwave Communications - Spring 2019 Lecture 8 2

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Linear Lightwave Demodulation - component level

Lightwave Signal Local

• scillator

90-degree hybrid coupler s(t)=sI(t)+sQ(t) sLO(t) z11 z12 z21 z22

+ _ Photodetector + _ Photodetector Photodetector Photodetector

iI(t) iQ(t) Electrical Signals

Figure: A balanced photodetector for the demodulation of the in-phase and quadrature components based on a 90-degree hybrid coupler.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 3

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Linear Lightwave Demodulation - functional level

+

• cos(2πfct)

−90o

sQ(t) sI(t) sI(t) cos(2πfct) −sQ(t) sin(2πfct)

+

Baseband Signal for the I-component Baseband Signal for the Q-component

sI(t) + isQ(t)

Complex Baseband Signal

2

Figure: Functional block diagram of a phase-synchronous demodulator.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 4

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Heterodyne Demodulation Block Diagram

(a) fIF B (b) −fIF −fc −fc fc fc

Lightwave signal before noise filtering Electrical signal after heterodyne demodulation

n(t) Optical Electrical

Lightwave signal after noise filtering Electrical Demodulation

• r(t)

r(t) = rI(t) + irQ(t)

• s(t)

sLO(t) Balanced Photodetector s(t) + no(t)

Figure: Functional block diagram of phase-synchronous heterodyne demodulation of a lightwave

• signal. (a) Block diagram of a lightwave passband demodulator.

(b) Signal and noise spectra after each stage along with the passband signal bandwidth B.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 5

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Heterodyne Demodulation Block Diagram

Using (??) with a = s(t) + no(t) ei2πfct and b = ALOei2πfLOt with fLO = fc − fIF, the passband electrical waveform r(t) can be written as (cf. (??))

• r(t)

= ALORe h(t) ⊛ sin(t) + no(t) ei2πfIFt (1) = ALORe s(t) + no(t) ei2πfIFt (2) = ALOs(t) cos 2πfIFt + φs(t) +ALO nIF(t), (3) where the passband noise process nIF(t) = Re[no(t)ei2πfIFt] is centered at the intermediate frequency fIF = fc − fLO Additional noise filtering, now in the electrical domain, may be used on this passband electrical waveform Show in Chapter 9 that for additive white gaussian noise, the impulse response of the detection filter that maximizes the sample signal-to-noise ratio, called a matched filter is a time-reversed replica of the received pulse waveform

ECE243b Lightwave Communications - Spring 2019 Lecture 8 6

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Output Pulse Energy used for Symbol Detection

The peak output of a matched filter to an input pulse is 2Ep/N0

Ep in the electrical pulse energy (not bit energy yet.) N0 is the power density spectrum of the white noise

The energy Ep in the passband electrical pulse is (cf. Chapter 2) Ep =

∞

−∞

• r2(t)dt

= 4rLO

∞

−∞ 1 2 s2(t) cos2

2πfIFt + φs(t) dt = 2rLOWp (4) = 2rLOeWp (5)

rLO = A2

LO/2 is the photodetected electrical signal generated from the lightwave

local oscillator signal, rapidly-varying cosine carrier averages to one half Wp = 1

2

∞

−∞ s2(t)dt is the directly-photodetected photocharge in the received

lightwave pulse s(t). Wp = Wp/e number of photoelectrons

Express using photoelectrons to make including shot noise easier

ECE243b Lightwave Communications - Spring 2019 Lecture 8 7

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Noise in Output Sample

The noise term N0 includes:

additive lightwave noise, shot noise, additive electrical noise

When the bandwidth of the noise filter is large, statistical properties of nIF(t) are the same as the incident spontaneous emission noise before optical noise-limiting filter

true when the local oscillator is deterministic

Consequently, the electrical-noise power density spectrum N0 caused by spontaneous emission is proportional to the lightwave-noise power density spectrum Nsp after the noise filter

ECE243b Lightwave Communications - Spring 2019 Lecture 8 8

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Noise Term - cont.

For a photodetector with a responsivity R, the electrical-noise power density spectrum N0 can be written in several equivalent forms N0 = 2R2PLONsp = 2erLOWn, (6) where RNsp = (ηe/hf)Nsp = ηeNsp = eWn, and Wn = ηNsp Nsp is the expected number of noise photons, and Wn is the expected number of noise photoelectrons generated during photodetection

ECE243b Lightwave Communications - Spring 2019 Lecture 8 9

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Shot Noise in Sample

The amount of shot noise depends on the total incident lightwave power, and on the local oscillator power When the LO power ≫ signal power or ≫ additive spontaneous emission power:

shot noise can be modeled is stationary, independent of the incident lightwave signal, accurately modeled as an additive white gaussian noise process with an electrical power density spectrum (Chapter 6) Nshot = 2erLO. (7)

The total noise power density spectrum N0 when both spontaneous emission noise and shot noise are present is N0 = Nshot + Nspe = 2erLO (1 + Wn) . (8)

ECE243b Lightwave Communications - Spring 2019 Lecture 8 10

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Spontaneous Emission Noise in the Sample

The power density spectrum from the spontaneous emission defined in (6) is written as Nspe . = 2erLOWn where Wn is the expected number of directly-detected photoelectrons generated by spontaneous emission If Nshot = 0, then (8) reduces to (6) and Nspe = N0 Additional thermal noise added after photodetection is typically small compared to the noise due to spontaneous emission

ECE243b Lightwave Communications - Spring 2019 Lecture 8 11

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Sample Statistic for a Matched Filter

.The electrical pulse energy Ep (4) and (8) for the electrical noise power density spectrum N0 gives Ep N0 = Ep Nspe + Nshot = 2erLOWp 2erLOWn + 2erLO = Wp Wn + 1 (9) This expression is valid for both a modulated real-baseband signal and a modulated complex-baseband signal

ECE243b Lightwave Communications - Spring 2019 Lecture 8 12

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Shot Noise Limit for Heterodyne Demodulation

For shot-noise-limited heterodyne demodulation with Wn much small that one, (9) reduces to Ep N0 ≈ Ep Nshot = Wp. (10)

This expression corresponds to shot-noise-limited power density spectrum Nshot equal to

• ne photon

A matched filter produces the largest value of Ep/N0 when the constant local

• scillator signal level is much larger than the incident signal level

A matched filter is not optimal for shot noise generated by the incident signal This case is discussed later

ECE243b Lightwave Communications - Spring 2019 Lecture 8 13

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Homodyne Demodulation of One Signal Component

The demodulated signal for the homodyne demodulation of one lightwave signal component to a real-baseband signal is determined by setting fIF = 0 and φs = 0 The electrical energy Ep in the demodulated real-baseband pulse r(t) = ALOs(t) is Ep = 4rLO

∞

−∞ 1 2 s2(t)dt

= 4erLOWp, (11) which is twice as large as the electrical energy in the passband pulse generated using heterodyne demodulation.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 14

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Homodyne Noise Terms

The electrical power density spectrum of the demodulated spontaneous emission is Nspe = 4erLOWn

both the signal and the noise are demodulated and filtered in the same way. The magnitude of the shot noise does not change compared to heterodyne demodulation Reason is the shot noise generated in the photodetection process depends solely on the power in the local oscillator, and not on the demodulated electrical waveform under strong LO assumption

Forming Ep/N0 gives Ep N0 = Ep Nspe + Nshot = 4erLOWp 4erLOWn + 2erLO = 2Wp 2Wn + 1 . (12)

ECE243b Lightwave Communications - Spring 2019 Lecture 8 15

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Shot Noise Limit for Homodyne Demodulation of One Component

For shot-noise-limited homodyne demodulation of one signal component, (12) reduces to Ep N0 ≈ Ep Nshot = 2Wp. (13) The factor of two in this expression corresponds to a shot-noise power density spectrum Nshot = 1/2

Equal to one-half a photon

This is the quantum-noise limit is achieved for the phase-sensitive homodyne demodulation of one signal component.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 16

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Homodyne Demodulation to a Complex-Baseband Signal (2 components)

W cos(fLOt) sin(fLOt) In-phase Quadrature Filtered Lightwave Signals

(b) (a)

Optical Electrical

B

rI(t) rQ(t) rI(t) + irQ(t)

2 sLO(t)

Balanced Photodetector

s(t) + no(t)

Figure: (a) Functional block diagram of direct-to-baseband homodyne demodulation. (b) Signal spectra for the I and Q signal paths. Left: the signal spectrum for each signal component before

• mixing. Right: the baseband signal components.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 17

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Demodulated In-Phase and Quadrature Components

The demodulated in-phase and quadrature components at baseband are rI(t) =

1 2 ALO Re

s(t) + no(t) rQ(t) =

1 2 ALO Im

s(t) + no(t) . (14) The complex-baseband waveform consists of the two real waveforms after demodulation that are jointly regarded as one complex waveform. This is r(t) = rI(t) + irQ(t) =

1 2 ALO

• h(t) ⊛ sin(t) + no(t)

eiθ(t) =

1 2 ALO

• s(t) + no(t)

=

1 2 ALO

• sI(t) + noI(t) + i

sQ(t) + noQ(t) . (15) Signal term Wp and the noise term Wn in (12) are halved as compared to a homodyne demodulator designed for one signal component The shot noise in each signal path does not change.

equivalent to heterodyne demodulator.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 18

Lecture 8 Demodulation Heterodyne Demodula- tion

Output Sample Shot Noise in Output Sample

Sample Statistic

Shot Noise Limit

Homodyne Demodula- tion

One Signal Component Shot Noise Limit Two Signal Components Shot Noise for Direct Conversion

Shot Noise for One and Two Signal Components

Shot noise for heterodyne demodulation direct-to-baseband homodyne demodulationis twice as large as homodyne demodulation of one signal component to a real-baseband signal Caused by the number of modes that contribute to the shot noise in the demodulated electrical signal For heterodyne demodulation, the frequency difference between the carrier and the local oscillator produces an additional image mode Mixing of the signal mode with the image mode produces a noise power density spectrum Nshot equal to one in accordance with (10) Image mode is not present in the homodyne demodulation of one signal component because the frequency of the carrier is equal to the frequency of the local oscillator Additional mode in direct-to-baseband demodulation is caused by the initial splitting of the signal.

ECE243b Lightwave Communications - Spring 2019 Lecture 8 19