noise analysis report hashemi

7/30/2019 Noise Analysis Report Hashemi

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Noise Analysis of Super-regenerative Receiver

Arka MajumdarElectronics and Electrical Communication Engineering

Indian Institute of Technology

Kharagpur,India

Email: arka [email protected]

Hossein HashemiUniversity of Southern California

Electrical and Electro-physics Department

Los Angeles,California, USA

Email: [email protected]

Abstract— In this paper, the noise in super-regenerative re-ceiver working in linear mode is analyzed both in time andfrequency domain. The effect of detection scheme for AM signalis included in the noise-analysis. The results obtained from thesetwo methods matches with the simulation results. These methodscan be used for any time-varying linear system.

I. INTRODUCTION

Super-regenerative receiver receives signal exploiting super-

regenerative oscillation principle [1]. Conductance of the R-L-C tank of the oscillator is periodically made negative as well

as positive by means of a periodic quench signal to make it

work sometimes as an oscillator and sometimes as a high-

Q filter. The oscillator is biased such that it can distinguish

between signal and noise during start-up. The receiver works

in linear mode when its oscillation is not allowed to reach the

stable value. It can distinguish between amplitudes and hence

primarily used for OOK modulation. But due to very high

sensitivity it can be used for AM signal also. The detection

scheme for AM signal is envelope-detection followed by peak

amplitude detection and reconstruction of the input signal

by sampling and holding the output envelope. The sampling

frequency is same as the frequency of the quench signal, whichshould be more than twice the band-width of the input signal

to avoid aliasing.

Noise in receivers is a major issue. [5] analyzes the noise in

super-regenerative receiver in frequency domain by modeling

it as a sampling receiver. In this paper, the system is modeled

as a linear time-varying system and noise-performance is

analyzed taking the detection scheme into account.

I I . TIM E-DOMAIN ANALYSIS

Response of an oscillator will depend on the injection time

of the impulses as the system is time-varying. The differential

equation describing the output voltage of the oscillator (com-

prising of L, C and a time varying conductance G(t)) (fig.1)

having an impulse at τ as input is

C dV

dt+ G(t)V +

1

L

V dt = qδ (t− τ ) (1)

where q is the charge content of the impulse and G(t) is

the time-dependent conductance. This change in conductance

is accomplished by a periodic quench signal of period Γ.

Here the quench applied is a sawtooth wave. The negative

conductance is realized by means of a cross-coupled BJT

Fig. 1. Simple oscillator with variable resistance

pair. Its trans-conductance is proportional to current. Hence

conductance G(t) has an expression for one time-period of

the quench signal (fig. 2):

G(t) = G0 −Kt (2)

Solving (1) the time-varying impulse response (normalized

w.r.t. the charge content of the impulse) of the oscillator

obtained is

h(t, τ ) = A(t, τ ) cos(θ(t, τ ) + ω0(t− τ )).u(t− τ ) (3)

where

A(t, τ ) =1

Cω0exp(

1

2C (G0(τ − t) −

K

2(τ 2 − t2))) (4)

and

tan θ(t, τ ) =1

2C (G0 −Kτ )

ω0(5)

ω0 is the natural frequency of oscillation and u(t) is a unit-step

function. The detection scheme shows that the effect of noise

at the detection point is of utmost importance. Magnitude and

phase of the output at the detection point (i.e. where the quench

turns off) w.r.t. the time of injection of impulse is shown in

fig. 2. This response can be explained intuitively. The impulse

injected just at the point when conductance is zero, faces thetotal negative conductance and hence the effect is maximum.

The impulses appearing during the positive conductance re-

gion face positive conductance for some time and the noise

appearing during the negative conductance will not face the

total negative conductance. This argument is applicable for

the signal also. The final output-amplitude is dependent on

the signal amplitude at the point where conductance becomes

zero. Hence for better performance, input SNR at that point

should be increased.



0 50 100 150 2000

1

2

3

4

5

6x 10

11

τ (nsec)

m a g n i t u d e r e s p o n s e

−1

−0.5

0

0.5

1

1.5

2

conductance (mili−mho)

G=0

Quench(not in scale)

conductancemagnituderesponse

94 95 96 97 98 99

0

1

2

3

4

5

6

τ (nsec)

p h a s e

a n g l e

( i n

r a d i a n )

Fig. 2. The magnitude and phase response at the detection point of thetime-varying network

Noise can be assumed to be sum of several impulses with

different amplitude [2].

n(t) =τ

(i(τ )).δ (t− τ ) (6)

As the system is linear, applying superposition principle the

noise amplitude at time t will be

nout(t) =τ

h(t, τ ).i(τ ) (7)

Noise power P out(t) is given by

P out(t) = E (τ

h(t, τ ).i(τ ))2 (8)

As the noise considered here is uncorrelated white gaussian

hence

P out(t) = E (i2(τ ))τ

h2(t, τ ) (9)

E (i2(τ )) is the noise-power at temperature T due to resistance

R, i.e., kT R

. Again (9) gives the expression for a discrete

stochastic process. But the process considered here is a con-

tinuous one. So for constant thermal noise

P out(t) =kT

R

h2(t, τ )dτ (10)

In circuit, one major noise source will be the noise contributed

by the transistors, whose PSD is given by 4kTgm. This is

not white as it depends on varying trans-conductance gm. At

time τ , gm is 2Kτ . Hence E (i2(τ )) can be expressed as

4kT (2Kτ ). Here P out(t) is given by

P out(t) = 4kT (2K )

τh2(t, τ )dτ (11)

Noise power at the detection point is given by

P out(T ) = 4kT (2K )

τh2(T, τ )dτ (12)

III. FREQUENCY-DOMAIN ANALYSIS

This analysis is done as the analysis done for switched

capacitor networks [3] and for cyclo-stationary noise sources

[4]. Let the fourier transform of the input noise to the receiver

and the noisy output of the receiver be X (ω) and Y (ω)respectively. Then the input-output relation with quench period

Γ is:

Y (ω) =

n=+∞n=−∞

H n(ω)X (ω − n2π

Γ) (13)

The noise is white gaussian and hence has a constant PSD

S x(ω). Relation between S x(ω) and output PSD S y(ω) is

S y(ω) =n=+∞n=−∞

|H n(ω)|2S x(ω − n2π

Γ) (14)

Using Parseval’s relation

n=+∞n=−∞

|H n(ω)|2 =1

Γ

τ 0

|H (ω, τ )|2dτ (15)

it can be written that

S y(ω) = S x(ω)1

Γ

τ 0

|H (ω, τ )|2dτ (16)

where H (ω, τ ) is the fourier transform of h(t, τ ) as given by

(3).

IV. DEPENDENCE OF NOISE ON CIRCUIT PARAMETERS

The super-regenerative receiver can also be thought just as

a sampling receiver. A quench satisfying Nyquist criteria will

cause no aliasing in band-limited signal but noise will bealiased. Hence as quench frequency f q increases, the noise

power decreases due to less aliasing. In frequency domain the

noise can be broken into two parts. Assuming that the noise

is passed through a band-pass filter (with bandwidthf q2

) the

noise power is calculated to get the first part. Here the effect

of aliasing is not taken into account. Then W f q

is added to the

first part to incorporate the aliasing effect, where W is the

system bandwidth [5].

The sampling effect is implicitly taken into account in time

domain in the term K . It can be shown that noise power is

a decreasing function of K which is proportional to quench

frequency.

With very high quench frequency the oscillator does notget enough time for oscillation building-up. [1] states that

one quench period should include a large number of natural

oscillation periods.

Apart from this, Noise-power decreases with increasing Q

of the tank ,i.e., increasing capacitance or resistance, while the

resonant frequency is kept constant by adjusting the inductor.

But these modifications also affect the signal performance.

Increasing capacitance increases the selectivity of the circuit

but the sensitivity of the circuit decreases. Hence, during



designing circuits all these considerations regarding faithful

representation of signal as well as noise reduction are to be

taken care of.

V. SIMULATION PROCEDURE AND RESULT

The theory of noise analysis is verified by modeling the

circuit of oscillator as shown in the fig. 3 in simulink. The

Fig. 3. Circuit for regenerative oscillator modeled in simulink

detection scheme (fig. 4) is also modeled. Exact BJT model is

Fig. 4. Block-diagram describing the detection scheme

incorporated. The tail current source is realized by a velocity-

saturated MOS. Fig. 5 shows the input and output of the

receiver used for AM signal. For noise analysis, random white

gaussian noise is passed through the system for several times

and the output squared is averaged to get the noise power intime domain. Fig. 6,7 show the noise-power obtained from

simulation and the theoretically predicted noise-power w.r.t.

time for both the resistor noise and gm noise. Fig. 8 shows

the PSD of the oscillator output as predicted by frequency-

domain analysis as well as obtained from the simulation. The

effect of detection is not included here. Fig. 9 shows the PSD

of the final detected output and the fourier transform of the

transfer function h(T, τ ) as shown in fig. 2.

The noise-power increases if the Q of the tank is in-

creased. Noise-power decreases with increasing tank capac-

itance. Noise power at the detection point obtained from

the time-domain analysis as well as simulation is plotted

against capacitances of the tank in the fig. 10. Fig. 11 showsnoise-power at the detection point w.r.t. quench frequency.

For building up oscillations some time is required. As the

quench frequency increases, duration of negative conductance

decreases and at high frequencies oscillations do not even

start up. So at frequencies higher than 6MHz the noise-

power becomes almost same, as oscillations do not even

build up. Assuming that the super-regenerative receiver is a

sampling receiver, the noise power can be obtained. The noise

is first passed through band-pass filter to get the aliasing-free

0 0.5 1 1.5 2 2.5 3

x 10−6

−1

0

1x 10

−5

0 0.5 1 1.5 2 2.5 3

x 10−6

−2

0

2x 10

−3

0 0.5 1 1.5 2 2.5 3

x 10−6

−0.01

0

0.01

0 0.5 1 1.5 2 2.5 3

x 10−6

0

2

4x 10−3

0 0.5 1 1.5 2 2.5 3

x 10−6

0

2

4x 10

−3

(a)

(b)

(c)

(d)

(e)

0 0.5 1 1.5 2 2.5 3

x 10−6

0

2

4x 10

−3

0 0.5 1 1.5 2 2.5 3

x 10−6

−0.02

0

0.02

time

(f)

(g)

Fig. 5. Working of regenerative receiver (without noise): (a)base bandsignal (b)quench signal (c)oscillator output (d)detected envelope (e)sample-hold output; (with noise): (f)sample-hold output (g)oscillator output (Quenchfrequency = 6MHz ; input signal frequency = 1MHz ; ω0 = 1.57 ×

1010rad/sec; C = 4 pF ; peak Quench current= 300µA); 1/G0 = 200Ω

TABLE I

AVERAGE NOISE POWER OBTAINED FROM DIFFERENT METHODS (INPUT

NOISE PS D IS CAUSED BY 50Ω RESISTANCE AND gm NOISE AT

TEMPERATURE 300K )

Time-domain Frequency domain Frequency domain

(by eqn.16) (aliasing effect)simulation -65.7576 dB -67.7583 dB -67 dB

theory -66.34 dB -67 dB -66.3 dB

noise and then the correction factor due to aliasing is added.

Table I shows the total noise-power obtained from these three

methods and the same as obtained from three different types

of simulations.

VI . CONCLUSION

Noise-analysis of super-regenerative receiver is done. The

result obtained from time-domain and frequency-domain meth-

ods matches largely with the simulation result. The detection

scheme is also taken into account.

ACKNOWLEDGMENT

We are highly grateful to Mr. Harish Krishnaswami and Mr.

Ankush Goel for many useful suggestions.

REFERENCES

[1] J.R.Whitehead,Super-Regenerative Receivers. Cambridge, U.K. : Cam-bridge University Press,1950



0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

4x 10

−6

time(nsec)

n o s i e a m p l i t u d e s q u a r e d

simulationtheory

Quench signal(not in scale)

Fig. 6. Comparison of noise power as obtained from simulation andtheory(constant thermal noise)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time(nsec)

a m p l i t u d e s q u a r e d

simulationtheory

Quench signal(not in scale)

Fig. 7. Comparison of noise power as obtained from simulation andtheory(time-varying gm noise)

[2] A.Hajimiri and T.H.Lee,“A General Theory of Phase Noise in ElectricalOscillators,” IEEE Trans.on Circuits and Systems-I , vol.50, pp.865-876,Jul.2003

[3] O.Oliaei,“Numerical Algorithm for Noise Analysis of Switched-Capacitor Networks,” IEEE J.Solid State Circuits, vol.36, pp.440-451,March 2001

[4] M.Okumara et. al.,“Numerical Noise Analysis for Nonlinear Circuitswith a Periodic Large Signal Excitation Including Cyclostationary NoiseSources ,” IEEE Trans.on Circuits and Systems-I , vol.40, pp.581-590,Sept.1993

[5] A.Vouilloz et. al.,“A Low-Power CMOS Super-Regenerative Receiver at1 GHz ,” IEEE J.Solid State Circuits, vol.36, pp.440-451, March 2001

Fig. 8. PSD as obtained from simulation and theory

−100 0 100 200 300 400 500 600 700 800

−120

−100

−80

−60

−40

−20

freq(MHz)

P S D ( d B )

simulationtheory

Fig. 9. PSD as obtained from simulation and theory considering detectionscheme

2 2.5 3 3.5 40

1

2

3

4

5

6

7x 10

−5

capacitance(pF)

a m p l i t u d e a t t h e d e t e c t i o n

p o i n t s q u a r e d

simulationtheory

Fig. 10. Variation of noise power at the detection point with tank capacitance

3 3.5 4 4.5 5 5.5 6 6.5 70

0.5

1

1.5

2

2.5x 10

−6

quench frequency (MHz)

a m p l i t u d e a t t h e d e t e c t i o n

p o i n t s q u a r e d

simulationtheory

Fig. 11. Variation of noise power at the detection point with quench frequency

noise analysis report hashemi

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