Time Domain

Every signal s(t) can be decomposed into a “spectrum” ofδ-pulses with the weighting functions(τ) as shown in eq. (5.5). The integral in eq. (5.5) is also called aconvolution integral with s(t) convoluted withδ(t) (or vice versa).

s(t) = If the response of a linear, time invariant system to a δ-pulse is h(t), the system output to s(t) can be described by

g(t) = We denote theδpulse response of the filter byh(t); the correspondingweighting function is defined by w(τ) = h(t−τ) and is a time reflected copy of h(t) shifted by the time of observation t. Due to their related nature both terms are often used as synonyms in literature. We will try to notify the difference by the namingh(t) resp.w(τ).

Eq. (5.4) should be compared to the convolution integral (5.6); the Fourier transforma-tion transfers a convolutransforma-tion in the time domain into a simple product in the frequency domain.

We consider in the following currentinput signals s(t) andvoltage output signalsg(t) -thus the unit of h(t) becomes F⁻¹. The integral is carried out between t =−∞ and

∞; however, for reasons of causality h(t) = 0 for t <0; in a real system also h(t) → 0 fort→ ∞holds.

Fig. 5.1 illustrates equation (5.6) and the meaning of h(t); we have added the noise sources found in a typical capacitive detector system; noise is introduced into the sys-tem due to three mechanisms (see also chapt. 4): parallel noise arises due to detector leakage shot-noise i²_sn and R_p’s thermal noise (R_p summarizes all parallel resistors and the amplifier’s parallel noise current source), serial noise is caused by the thermal noise of R_s (equivalent to amplifier’s serial noise voltage according to eq. (4.29)).

Shot-Noise in the Time Domain

Devices with a voltage barrier like diodes behave as sources of current noise. In a diode the barrier allows current pulses in one direction with a Poisson distribution in time.

The derivation following is an illustrative interpretation of Carson’s theorem [Ziel70].

A charge δq applied to the circuit at time τ produces an output voltage of δqh(t−τ) at time t where h(t−τ) describes the current-to-voltage response of the circuit (the input impedance is assumed to be 0 Ω). The output of the shot noise s(τ) source can be divided into time intervals of length δτ, each with charge δq. To receive the total output voltage at time t, all the separate output voltage contributions δq produced by charges in the individual time intervals have to be summed. Of courseδq differs for each interval.

To get an measure of the fluctuations of δq, it is assumed that within each interval [τ, τ+δτ] there are many charge pulses of size e (electron charge). There is a mean rate

current-to-voltage filter h(t) noisy

resistors R_p

Cin

R_s

g(t) isn

s( )τ 2

Figure 5.1: Filter with current-voltage transfer function h(t); parallel noise arises due to detector leakage shot-noise i²_sn and R_p’s thermal noise, serial noise is caused by the thermal noise of R_s.

δq)

<δq>, var(

δq

q>

<δ δqh(t-τ)

δτ τ+ δτ τ

= ne²δτ e

g(t)

t

= neδτ δq)

var(

τ s( )

Figure 5.2: Shot noise originates from discrete charge carriers passing a “barrier” like a diode junction. The noise is viewed using a filter characterized by h(t) resp.H(ω).

of n electrons/second so there will be an average of nδτ pulses per interval [τ, τ +δτ];

the average charge in the interval will beneδτ.

The mean voltage at the ouput is given by integrating the mean charges with the ap-propriate weighting (with ne=I₀ mean current):

g(t) =

Eq. (5.9) describes the filter output voltage g(t) at time t of a system switched on att₀ (the upper integration limit can be set to ∞ since h(t) = 0 fort <0. Hence the circuit behaviour can also be studied during the “warm up” of a circuit. For the stationary case, we have to sett0=−∞(in practice it would suffice to go back by ∆twithh(t) = 0 fort >∆t).

However, we are more interested into the variance (varg)(t). From Poisson statistics follows for the variance of δq

var (δq) =ne²δτ . (5.8)

By “propagating” the input variance to the output of the filter (taking into account that the variance is transformed by the square of the transfer function) we receive

(varg) (t) = (σ²g)(t) =

Frequently the integral (5.9) is expressed in the simplified but less instructive form by rearrangement of the integration range

(σ²g)(t) = 1

To cross check eq. (5.9) with the value expected from frequency domain calculation we assume the stationary case setting the upper integration limit to ∞:

(σ²g)(∞) :=σ²g= 1 2(2eI₀)

Z∞

−∞

h²(τ)dτ (5.11)

By employing twice the inverse Fourier transformation (see sect. 5.1.1) h(t) = 1

2π Z∞

−∞

H(ω)e^jωtdω (5.12)

and going over to the unilateral spectral noise density we obtain

σ²g= 1

I = ne0 I = ne0

q>

<δ = 0 δq) var(

R R

= 2ne²δτ

Figure 5.3: A (parallel) noisy resistor is modelled by a noiseless resistor with two an-tiparallel current sources with equal mean currents. The variance of charge per time slot [τ, τ +δτ] of each current source is var(δq) =ne²δτ.

which is also known as Parseval’s theorem [BS88]. Eq. (5.13) is exactly what we expect from the frequency domain calculation when feeding shot noise given by

σ²i=i²−I₀² = 2qI₀∆ν [in A²] (5.14) into a filter with the current-voltage transfer function H(ω).

In order to calculate the response of a certain time invariant system to a noise source at the input, both equations (5.7) and (5.13) can be used.

Thermal Noise in the Time Domain

Thermal noise (see also sect. 4.1) is caused by the thermal motion of charge carriers in conductors/resistors. For the calculation of the system output noise in the time domain when connecting a resistor to the filter input, we apply the following model:

The thermal noise of a resistor R_p is modelled by two antiparallel current sources in parallel to the noiseless resistor, which have equal mean currents; the variance of charge per time slot [τ, τ +δτ] of each current source I0 is var(δqi) = ne²δτ. Thus, the total variance of both current sources adds up to

var(δq) = 2ne²δτ . (5.15)

The problem now is, that - since there is no DC current observable - we cannot determine the mean rate electrons/second n. By investigations on the thermal motion of charge carriers using statistical thermodynamics [John28, Nyq28] it can be shown that

n= kT

Re² (5.16)

k= 1.38·10²³J/K,T absolute temperature Rresistance,e= 1.6·10⁻¹⁹ electron charge

and the variance at the filter output becomes (varg)(t) = (σ²g)(t) =

Eq. (5.17) describes the noise for a parallel input resistor; in principle the resistor would have to be included in the weighting function h(t); however, for a zero Ohm filter input impedance the resistor does not have any impact on the transfer characteristic h(t).

For completeness the noise calculated with eq. (5.17) is cross checked with the noise from frequency domain study. We assume again the stationary case and rearrange the integration range as in eq. (5.10):

(σ²g)(t) = 1

In the stationary case eq. (5.18) becomes σ²g= 1

and from employing twice the inverse Fourier transformation we obtain σ²g= 1 Eq. (5.20) thus is compatible with the Nyquist-formula (4.3)

σ²v=v² = 4kT R∆ν [in V²] resp. σ²i=i²= 4kT

R ∆ν [in A²] .

In order to calculate the noise arising from a resistor R_s in series to the input of a network, we transfer the two current sources to two (noisy) antiparallel voltage sources of value V₀=R_sI₀=R_sne withnas defined in eq. (5.16) (see fig. 5.4).

The average of the two voltages cancels, but the variance of the voltage-time products per time slot becomes

R²_svar(δq) = 2R_s²ne²δτ . (5.21) Equation (5.21) is the equivalent of eq. (5.15) and gives the variance in terms of resistance

×charge resp. voltage×time which serves mathematically as input to a voltage-voltage transfer function.

Unfortunately, we only know the filter response to a (parallel) current input; therefore we have to calculate the system response for a voltage applied in series to the filter input. Taking into account the input capacitance C_in, a serial voltage sourceV_s causes a current

iin=CindVs

dt (5.22)

R

Figure 5.4: A (serial) noisy resistor is modelled by a noiseless resistor with two antiparal-lel voltage sources in series, which have equal mean voltages. The variance of the voltage-time products per voltage-time slot [τ, τ +δτ] of each voltage source is R²_svar(δq) =R_s²ne²δτ. to flow into the “filter” h(t) (neglecting the impact of R_s). In other words, a serial δ-voltage pulse causes a filter input current ofC_inδ(t).˙

For an arbitrary voltage Vs in series with the filter input we obtain for the voltage at the system output generally true for real system).

Eq. (5.23) can be interpreted in such a way that C_inh(t) is the˙ voltage-voltage-transfer function of a filter characterized by a current-voltage transfer function h(t) with a ca-pacitive input load C_in.

How doesR²_svar(δq) propagate to the filter output ? Again we integrate over all relevant

“voltage-time” products R²_svar(δq) betweent₀ and t (resp.∞):

(varg)(t) = (σ²g)(t) = Z∞

t0

C_in² 2R²_sne²h˙²(t−τ)dτ (5.24)

Substituting nfrom eq. (5.16) yields

(varg)(t) = (σ²g)(t) = 1

2C_in² (4kT R_s) Z∞

t0

h˙²(t−τ)dτ . (5.25)

Rearrangement of the integration range yields (σ²g)(t) = 1

and in the stationary case

σ²g= 1

2C_in² (4kT R_s) Z∞

−∞

h˙²(τ)dτ . (5.27)

Refering to the discussion of the charge amplifier with series resistance in sect. 4.4.3 it becomes immediately clear that we can use formula (5.27) also for the description of the amplifier’s inherent serial noise. The equivalent noise resistance is related to the input transistor’s transconductance g_m via

R_s,eq= 2

3g_m . (5.28)

To compare the result achieved in eq. (5.27) to a frequency domain analysis we have to take the derivative of the Fourier transform

dh

and substitute it into eq. (5.27) yielding (after transition to the unilateral spectral noise density)

With the Fourier transform of eq. (5.22)

I_in(ω) =jωC_inV_s(ω) (5.31)

and defining from this the voltage-voltage frequency response of the filter under consid-eration

H^◦(ω) :=jωC_inH(ω) (5.32)

we obtain from eq. (5.30) σ²g= 1

2π(4kT R_s) Z∞

0

|H^◦(ω)|²dω= (4kT R_s) Z∞

0

|H^◦(2πν)|²dν . (5.33)

Eq. (5.33) describes the voltage noise at the filter output due to an input voltage noise σ²v= 4kT R_s∆ν, thus confirming our result from time domain calculation.

It should be noted again that the time domain eqs. (5.9), (5.17), and (5.25) are more general than the frequency domain equations obtained by application of Parseval’s theo-rem since they cover nonstationary processes as well. The 1/f-noise (sect. 4.1), however, can only be handled for stationary processes.

Im Dokument A CMOS Mixed-Signal Readout Chip for the (Seite 61-68)