Loop analysis of adaptive notch filters

: Adaptive notch filters have been the focus of intense research for more than three decades. Low computational requirements and good performance make them attractive for tracking frequency modulated signals. Despite the extensive literature on adaptive notch filters, algorithm extensions and new models for describing the dynamics of the notch frequency continue to be proposed. In this study, the equivalence between adaptive notch filters using a plain gradient (PG) algorithm and frequency lock loops (FLLs) with exponential filtering is established. FLL theory is then used to analyse the noise performance and signal tracking capabilities of adaptive notch filters. A linear model describing the dynamics of the filter adaptation process is derived and the concepts of loop and Doppler bandwidths are introduced. Criteria based on the loop and Doppler bandwidths are suggested to set the PG adaptation step. Finally, algorithm extensions based on the FLL theory are proposed. Theoretical results are supported by Monte Carlo simulations which show the validity of the analysis performed.


Introduction
Adaptive notch filters have been studied for more than three decades. More specifically, the infinite impulse response (IIR) adaptive notch filter with constrained poles and zeros suggested by Nehorai [1]h a s been the focus of intense research for its low computational requirements and good performance. Adaptive notch filters are widely used for removing instantaneously narrow band interference [2] such as unwanted continuous waves and chirp signals [3]. For example, they find application in biomedical signal processing [4] and direct sequence spread spectrum communications [5]. In Global Navigation Satellite Systems (GNSS), the adaptive notch filter has been adopted for interference removal [6,7]f o ri t sa b i l i t yo f attenuating only a narrow part of the spectrum, thus preserving most of the wideband Galileo/GPS signals.
Despite the extensive literature on the subject, algorithm extensions and new models for describing the dynamics of the adaptive notch filter continue to be proposed. For example, Punchalard et al. [8] suggested a modified sign algorithm (SA) to be used in conjunction with the filter structure proposed by Nehorai [1]. In [9], new error criteria were introduced for adapting the notch filter parameters. Loetwassana et al. [10,11] suggested a modified plain gradient (PG) algorithm to avoid biases in the parameters estimated by a second-order adaptive notch filter. These are just a few examples of recent developments on adaptive notch filters.
The steady-state error of the adaptive notch filter has been mainly studied using the mean-square-error criterion [11,12] and the linear filter approximation (LFA) approach [13,14] has been developed for characterising the filter dynamics. In the LFA, two transfer functions, defining two equivalent filters, are introduced and the frequency estimate provided by the adaptive algorithm of the notch filter is described as the linear combination of filtered noise and filtered input signal frequency.
The LFA has been recently extended to generalised adaptive notch filters by Niedzwiecki and Kaczmarek [15] whereas a linear approximation accounting for the presence of a frequency acceleration in the input signal frequency has been considered by Niedzwiecki and Meller [16].
In this paper, a new approach is proposed for the analysis of adaptive notch filters. In particular, the equivalence between adaptive notch filters and standard frequency lock loops (FLLs) [17,18]i sfirst shown. The equivalence is valid when the PG algorithm used to update the adaptive notch filter is designed to minimise the energy at the output of the moving average (MA) part of the notch filter. The equivalence is then extended to the general case that considers the minimisation of the energy of the notch filter output. In this case, the adaptive notch filter is equivalent to an FLL with exponential filtering [19,20].
FLL theory is then used to analyse the dynamics of the system. In particular, it is shown that the evolution of the frequency of the filter notch can be described using two equivalent filters as in the LFA approach. The noise transfer function found is equivalent to that obtained in [13,14]. However, a different parameterisation for the signal component is obtained. The linear approximation obtained using the FLL theory is used to introduce the concepts of loop and Doppler bandwidths which characterise the filter performance in terms of noise rejection and tracking capabilities. Criteria based on the loop and Doppler bandwidths are finally suggested for setting the adaptation step of the algorithm used for updating the notch frequency: in the previous literature, the adaptation step was selected using empirical approaches.
Finally, algorithm extensions based on FLL theory are discussed: the theory developed provides a unified analysis of the adaptive notch filter and its extensions.
Theoretical results are supported by Monte Carlo simulations that show the validity of the analysis developed. Simulations also illustrate the impact of loop and Doppler bandwidths on the design of the adaptive notch filter.
The remaining of this paper is organised as follows: the adaptive notch filter is introduced in Section 2. The equivalence with FLLs is demonstrated in Sections 3 and 4. The linear equivalent model describing the filter dynamics is derived in Section 5, whereas the concepts of loop and Doppler bandwidths are introduced in Section 6. Algorithms extensions are discussed in Section 7 and simulation results are provided in Section 8. Finally, Section 9 concludes the paper. equations governing the IIR notch filter detailed. Then, the filter zero, z 0 , is allowed to be time-varying leading to the adaptive notch filter.
The notch filter considered in this paper is characterised by the transfer function [6,21] where z 0 is the complex filter zero, which determines the frequency of the filter notch. k α is the pole contraction factor and assumes values in the range [0, 1). This restriction is applied in order to guarantee the stability of the filter. Equation (1) defines the inputoutput relationship where x[n] and y[n] are the samples taken at the input and output of the system, respectively. The filter zero, z 0 , is constrained to lie on the unit circle and thus it can be expressed as where f 0 is the frequency of the filter notch expressed in units of radians/second. In this paper, frequencies will be expressed in units of radians/second and conversion in hertz (Hz) can be obtained through scaling by 2π. T s is the sampling interval of the signal at the input of the filter. The output of the MA part of the filter is given by The final output of the filter, y[n], can be rewritten exploiting the equivalent representation of (1) In particular, y[n] can be expressed as where is the output of the auto regressive (AR) part of the filter. Equation (6) has the same functional form of (4) where x[n − 1] has been replaced by x r [n − 1], the output of the AR part of the filter. Equations (1)-(7) describe the basic structure of a time-invariant IIR notch filter. An adaptive version of such filter is obtained by allowing f 0 and z 0 to be time varying. In this case, an adaptation block is used to progressively update f 0 and z 0 . Since the adaptation block can update f 0 and z 0 at each time instant, the dependence on the time index, n, is explicitly added in the following.
The structure of the adaptive notch filter is detailed in Fig. 1: the adaptation block uses the input samples, x[n], to estimate the notch frequency, f 0 [n], and to adjust the filter zero, z 0 [n]. The adaptation block used to estimate the notch frequency is the main focus of this paper and is analysed in detail in the following sections. In particular, PG algorithms are considered for the estimation of f 0 [n]. When the PG approach is adopted, the filter zero, z 0 [n] is adapted using a gradient descent algorithm [22]. For example, Borio et al. and Calmettes et al. [6,23] adapted z 0 [n] by minimising the instantaneous energy of filter output (6) In [1,12], the energy of the filter output was averaged considering the past L output samples whereas Tichavsky and Handel [13] considered the minimisation of the energy filtered using a truncated exponential filter where ρ ∈ (0, 1] is the filter forgetting factor. Note that for L = 1, (9) and (10) degenerate to (8). In the following, the analysis is restricted to PG algorithms minimising (8). Cost function (8) has been selected for its simplicity: more complex cases such as (9) and (10)   Thus, when the gradient is computed, f 0 [n] is not available and the adaptation block can only use a past estimate of the notch frequency, f 0 [n − 1]. For this reason, the output of the MA part of the filter is not directly available for gradient computation and the adaptation block can only evaluate the approximatioñ Thus, the instantaneous energy at the output of the MA part of the notch filter is estimated by the adaptation block usingỹ m [n] instead of (11). Note that in the steady-state conditions, we have and the minimisation of the approximated cost function implies the minimisation of the actual one. Using (12), the instantaneous energy at the output of the MA part of the notch filter can be expressed as In the absence of noise, the input signal, x[n], can be expressed as and instantaneous power (13) becomes where is the instantaneous frequency of x [n]. PG algorithms update the notch filter frequency according to where μ is the adaptation step and g [n] is the stochastic gradient given by where the gradient has been computed using (15). Finally, the notch frequency is updated according to Equation (19) defines the structure provided in Fig. 2a. The input signal, x[n], is at first multiplied by a delayed and complex conjugated version of itself. This operation is equivalent to a phase differentiation on the input signal. After phase differentiation, the signal is multiplied by the complex conjugate of z 0 and the discriminator output is computed. The discriminator output corresponds to the opposite of gradient (18). The discriminator output is then used to update the frequency estimate, f 0 , and to adapt the notch filter zero. The diagram in Fig. 2a can be restated as in the bottom part of the figure which shows that the simple adaptive notch filter considered in this section is equivalent to a standard FLL [18,[24][25][26]. In Fig. 2b, , a delayed and complex conjugated version of the complex carrier obtaining the signal . Phase differentiation is then performed and the product is computed. Signal (22) is equal to the product at the input of the discriminator in Fig. 2a. The discriminator output is then computed and an additional integrator is used in Fig. 2b to evaluate i[n] and to close the loop. The analysis provided above shows that the adaptive notch filter using a PG algorithm minimising the energy at the output of the MA part of the filter is equivalent to a standard FLL of the first order. The gradient operator defines the loop discriminator and μ, the adaptation step, defines the loop filter. This equivalence will be exploited in Section 6 to analyse the tracking properties of the adaptive notch filter and provide a methodology for selecting the adaptation step, μ.

FLL with exponential filtering
When cost function (8) is considered, gradient (18) assumes the following form where x[n − 1] has been replaced by the output of the AR part of the filter, x r [n − 1]. An equivalent FLL model for the adaptive notch filter using a PG algorithm minimising the power at the filter output can be obtained by following the procedure introduced in Section 3. The product in brackets in (23) can be rewritten introducing a complex carrier (20) where the property has been exploited. The term is a baseband version of x r [n]. The recurrence equation governing the generation of x Br [n] is directly derived from (7) and it is given by Assuming that z 0 is slowly varying with time, the following approximation can be made It is noted that the time variations of z 0 are governed by (17): since the algorithm operates for small values of the adaptation step, μ, small variations of z 0 can be assumed. Using (29), recurrence (28) becomes Equations (30) and (25) allow the derivation of the equivalent FLL model for the adaptive notch filter with constrained pole and zero as shown in Fig. 3. The adaptive notch filter is equivalent to an FLL with a pre-filtering stage inserted before phase differentiation. Pre-filtering is implemented by h k [n], which is the impulse response of the exponential filter defined by recurrence (30). The transfer function associated to h k [n]i s Note that H k (z) is a low-pass filter with unit gain at the zero frequency. The bandwidth of the filter is regulated by k α which can also be interpreted as a forgetting factor that leads to sharper transfer functions as it approaches unity. For k α = 0, the standard FLL considered in Section 3 is obtained.

Equivalent model
In this section, an equivalent model for the FLL with exponential filtering is derived. The analysis follows the standard approach adopted for the characterisation of tracking loops [17,18] where an equivalent model is at first derived. For the analysis, signal model (14) is modified by considering the presence of a noise term, η[n], and by assuming that the signal amplitude is slowly varying with time, i.e. A[n] ≃ A. In this way, (14) becomes where η[n] is a white circularly symmetric complex Gaussian random process with independent and identically distributed real and imaginary parts. η[n] has total variance σ 2 .
The discriminator output in Fig. 3 can then be expressed as that is the sum of a useful and a noise components. The properties of signal and noise components in (35) are analysed separately in the following sections.

Signal component
In order to determine the signal component at the discriminator output, it is necessary to characterise s Br [n], which is obtained by The coefficients, h k [m], have been obtained by expanding the geometric series defined by transfer function (31). Using expansion (36), it is possible to compute the signal component at the discriminator output. In particular Assuming that the arguments inside the sine functions in (37) are small, then (37) can be linearised as Approximation (38) is based on the assumption that the algorithm has been properly initialised and thus that the residual phase term, Δj[n], is small for all time instants, n. In this way, differences inside the sine functions in (37) remain small for all values of m.
This assumption corresponds to the equilibrium state (ES) hypothesis introduced by Tichavsky and Handel [13], Tichavsky and Nehorai [14] for the derivation of the LFA. The proper initialisation of the adaptation algorithm can be performed using frequency acquisition techniques [18], for example based on the fast Fourier transform algorithm. Alternatively, the properties of the adaptive notch filter can be exploited and unaided frequency acquisition can be performed [18]. When the frequency of the filter notch is far from the frequency of the input signal, the adaptation algorithm progressively sweeps all possible frequencies until frequency lock is achieved. Thus, condition (38) is also valid when data have been processed for a long time and the algorithm is operating in the steady-state conditions. In particular, coefficients, h k [m], decay exponentially and only differences obtained for small values of m influence the summation in (37).
Since small values of m are considered and assuming that, in steady state, Δj[n] is affected only by small variations, differences in (37) remain close to zero, justifying (38). The step size of the adaptation algorithm influences the unaided frequency acquisition process which can be improved using variable step size algorithms [27]. The analysis of variable step size algorithms is outside the scope of this paper. where is the residual frequency error at the instant n. In this way, (38) becomes Coefficients g k [l ] assume the following form and define an exponential filter with a direct current (DC) gain different from 1. Finally where is the discriminator gain. From these considerations, it emerges that the signal component at the discriminator output is a scaled and filtered version of the residual frequency error, Δf [n].

Noise component
The noise component at the discriminator output is composed of two terms and The first term is due to the interaction between the noise and the signal components, whereas N[n] is a pure noise term. The two terms are uncorrelated. C[n] can be characterised by assuming that, in lock conditions, s B [n] is almost constant and Using this approximation, cross-product (45) can be expressed as From the analysis reported above, it emerges that the cross-product term is obtained by scaling and filtering a zero mean white Gaussian process with variance σ 2 /2. The filter transfer function can be derived from (48) and it is given by which is the baseband version of (5) and it is a combination of a numerical derivative and an exponential filter.
It is noted that this component can be neglected for a wide range of k α and of input signal-to-noise ratio (SNR) values where the SNR is defined as Theoretical results are supported by Monte Carlo simulations. In particular, it has been verified through simulations that for k α =0, the case of a standard FLL, noise (50) contributes to about 13.7% of the total noise power. When k α = 0.9, the contribution of N[n] is about 1.55%. As the SNR increases the contribution of N[n] further diminishes. For this reason, only the cross-product term will be considered.

Equivalent model
The results reported above allow one to derive the equivalent linear model depicted in Fig. 4 that is expressed in the signal phase/ frequency domain. In Fig. 4, an equivalent white noise term, N d [n] has been introduced. N d [n] is zero mean and has variance (53) The proof of (53) is provided in Appendix 2. After filtering, N d [n] becomes a coloured noise which models the cross-product term analysed in Section 5.2. The phase term, j 0 , is the argument of the local carrier, i[n] The model in Fig. 4 can be simplified by collecting terms common to the noise and signal branches thus obtaining the final loop depicted in Fig. 5. This model is equivalent to that obtained in [19,28] for tracking loops with exponential memory discriminators. Thus, the theory developed in [19,28] can be adopted for the analysis of the adaptive notch filter. When k α = 0, the equivalent FLL loop derived in [29] is obtained. From the model in Fig. 5, it is possible to show that the Z-transform of the notch filter frequency, f 0 [n], is given by the linear combination of filtered signal phase and noise is the transfer function of the numerically controlled oscillator model adopted in Fig. 5. Using (31), the transfer function multiplying the noise term in (8) can be written as which is equivalent to the noise transfer function obtained in [13,14] in the LFA. A different transfer function is obtained for the signal component. In (55), the final frequency estimate is expressed as a function of the phase of the input signal: the equivalence with FLLs provides an alternative to the LFA approach for the analysis of notch filters. From (57), it is possible to note that the term, μT s G d , has to be dimensionless. Thus, it is convenient to express μ as a function of the normalised update step Note that the signal before multiplication by μ is in units of radians times G d . After multiplication by m = ( m/T s G d ), f 0 [n] is obtained and is in units of radians/second.
The signal phase, j[n], is related to the signal frequency through (16) and thus (55) can be rewritten as a function of f x [n] Finally, (59) can be restated introducing normalised adaptation step (58) , the normalised frequency noise with variance (62) Equation (60) will be used in the following section to characterise the tracking properties of the adaptive notch filter.

Loop bandwidths and adaptation step
The two transfer functions, H nf (z) and H f (z), derived in Section 5 define the Doppler and loop bandwidths, respectively [29]. is the Doppler bandwidth expressed in Hz [29]. Using tables in [30] (p. 298), the Doppler bandwidth can be expressed as Using (65) and (62), the variance of f 0 [n] becomes Equation (66) is in agreement with the findings obtained by Tichavsky and Handel [13] using the LFA approach. Moreover, the square root of (66) corresponds to the Doppler tracking jitter derived in [29] for standard FLLs. The transfer function, H f (z), can be used to evaluate the loop bandwidth. Note that in standard tracking loops, the loop bandwidth is defined with respect to a noise transfer function obtained considering a different linear model for the loop representation [17,29]. In this case, H f (z) is used and the loop bandwidth characterises the tracking capabilities of the adaptive notch filter in the absence of noise. Noise performance is determined by the Doppler bandwidth introduced above. A discussion about the difference between loop and Doppler bandwidths can be found in [29].
The loop bandwidth is defined as and can be computed in closed form as [30] It is noted that for k α = 0, (68) becomes which is the standard expression for the bandwidth of first-order digital tracking loops [31] (Table IV). In this respect, the normalised adaptation step, m, can be interpreted as the coefficient defining the filter of a first-order digital tracking loop [25,31]. Thus, m can be obtained either by fixing the Doppler bandwidth and inverting (65) is respected, then f 0 [n] is slowly varying with time. In particular, (70) implies that f 0 [n] has only low frequency components. Condition (70) is usually required for the stability of digital tracking loops [31] and it is necessary in order to have smooth estimates of the input signal frequency and to cope with noise. This fact justifies the assumption of slowly varying processes and (29).

Algorithm extensions
The PG algorithm can be extended exploiting the equivalence with the FLL. In particular † Gradient (23) can be replaced using modified FLL discriminator functions. † The normalised adaptation step, m, can be replaced by integrator-based filters which can lead to loops of order higher than 1.
It has been shown that the discriminator output in the FLL equivalent model is obtained as a scaled version of the opposite of gradient (23). In particular, (35) was directly obtained by scaling (23). Alternative PG algorithms can be obtained by replacing 2ℑ· {}with a different function. Possible candidates are [25] Equations (71a)-(71c) are directly derived from FLL discriminators commonly used in the relevant literature [25] and have a gain, G d , which is independent from the amplitude of the input signal, A. The analysis developed in Section 5 showed that (35) led to gain (44), which is proportional to A 2 . This dependence has to be removed by normalising μ. However, A and A 2 are in general unknown and need to be estimated. The problem of amplitude dependency is well known in the literature and, for example [32] highlighted the need of an automatic gain control to maintain the signal amplitude within a known range and avoid instabilities in the adaptive notch filter. Normalised gradient algorithms were introduced in [19,33] to remove the amplitude dependency: A 2 can be determined using a reference signal [33] or employing a signal energy estimator as in normalised least mean squares algorithms [22]. A solution commonly adopted in the FLL literature [25] is the use of amplitude independent discriminators such as (71a)-(71c). Discriminator (71a) leads to the multi-frequency tracker considered in [13] when a single sinusoid is present. A form of SA [8,32] is obtained using (71c). From this discussion, it emerges that several adaptive algorithms employed for IIR notch filters can be obtained exploiting the FLL theory. The adaptive notch filter can be further extended by replacing the adaptation step, μ, by an integrator-based filter with transfer function where μ m are the filter coefficients. For K =1 (first-order loops), transfer function (72) reduces to a constant gain that corresponds to the adaptation step, μ. The integrator gains, m m K−1 m=0 , can be determined using the modified controlled-root formulation [31] for loops with exponential filtering [19].

Simulation results
In this section, simulation results supporting the theoretical findings described in Sections 5 and 6 are provided. In particular, the adaptive notch filter described in Section 2 has been tested in the presence of signal (32) for different SNR conditions. The tests have been conducted considering a sampling frequency f s =(1/T s ) = 10 MHz. For the noise performance analysis, a complex sinusoid with constant amplitude and constant frequency was generated at each simulation run. The frequency of the sinusoid was selected randomly and was changed at each simulation run. The sinusoid was corrupted by a white complex Gaussian noise the variance of which was selected according to the SNR considered for the experiment. Fig. 6a shows the standard deviation of f 0 [n] for a constant Doppler bandwidth, B d = 10 kHz, and for several values of k α .I n Fig. 6a, the standard deviation has been expressed in Hz in order to provide a more immediate understanding of the results. The curve denoted as 'theoretical' has been computed using (66) whereas the other curves have been obtained through Monte Carlo simulations: for each SNR value, signal (32) has been generated and processed using the adaptive notch filter. The frequency estimates, f 0 [n], provided by the PG algorithm have then been used to estimate the frequency standard deviation. For each SNR condition, N =10 7 frequency values have been used for estimating the standard deviation. In order to have a constant Doppler bandwidth, the normalised adaptation step, m, has been obtained by solving (65) From Fig. 6a, it emerges that the noise performance of the adaptive notch filter is essentially determined by the Doppler bandwidth and that the frequency standard deviation only marginally depends on k α .F o rk α > 0 there is a good agreement between (66) and Monte Carlo simulations. For k α = 0, simulation and theoretical results start deviating for SNR values lower than 15 dB. This result is to be expected since, for low SNR values and for small contraction factors, the contribution of noise (50) cannot be neglected as in (66).
The impact of the Doppler bandwidth is further analysed in Fig. 6b where k α has been set to 0.9 and several values of B d have been considered. A good agreement between theoretical and simulation results can be observed. For B d = 5 kHz, a small deviation between Monte Carlo and theoretical results can be observed. This is due to the fact that the theory developed in Section 4 assumed that f 0 [n] is slowly varying with time. As the Doppler bandwidth increases also the loop bandwidth increases and condition (70) is no longer respected.
The impact of the loop bandwidth is investigated in Fig. 7 that shows the evolution of the residual frequency error as a function of time and in the absence of noise. The residual frequency error is the difference between the signal frequency, f x [n], and f 0 [n]. In Fig. 7, simulated curves have been obtained using the adaptive notch filter to process a noiseless complex sinusoid with a constant frequency, f x = 3 kHz, whereas the linear approximations have been computed by processing a frequency step of amplitude f x with the filter defined by the signal transfer function, H f (z). The residual error quickly converges to zero and the convergence speed is mainly determined by the loop bandwidth. Although different contraction factors are considered in Fig. 7, the residual frequency errors show similar behaviour. This is due to the fact that, the loop bandwidth has been kept constant during the different experiments. In this case, the normalised adaptation step, m, has been selected according to In Fig. 7, a good agreement between simulations and the linear approximation developed in Section 5 is observed. These results support the validity of the theory developed in this paper. The impact of the loop bandwidth is further analysed in Fig. 8 that shows the convergence behaviour of the residual frequency error for several values of B n and a constant pole contraction factor, k α = 0.9. Large loop bandwidths improve the convergence speed of the algorithm. Also in this case, a good agreement between simulation and theoretical results can be observed.
Finally, the response of the notch filter adaptation algorithm is considered in Fig. 9 in the presence of noisy sinusoids. In particular, model (60) is additive with respect to the input signal and noise components since non-linear effects in discriminator output (35) are accounted for in the definition of the equivalent noise term. Thus, the model derived can also be used to analyse the behaviour of the adaptive notch filter in the presence of both signal and noise components. In Fig. 9, frequencies estimated by the adaptation algorithm are compared with the noiseless equivalent signal impulse response which effectively describes the average behaviour of the system. Moreover, most frequency estimates are within the band defined by the frequency of the input signal and the standard deviation obtained from (66). This confirms the ability of the proposed model to effectively capture the behaviour of the system. In Fig. 9, two cases are considered for different contraction factors and SNR conditions. The simulations have been performed for a constant Doppler bandwidth, B d =10kHz, and for a signal frequency equal to 100 kHz. In both cases, a good agreement between simulations and theoretical results is obtained.

Conclusions
In this paper, the equivalence between adaptive notch filters and FLLs using an exponential filter was established. In particular, the pole contraction factor of the notch filter is equivalent to the forgetting factor of the FLL exponential filter. The equivalence allows the usage of the FLL theory for analysing the dynamics of the adaptive notch filter. In particular, FLL theory was used to derive an equivalent linear model and to introduce the concepts of loop and Doppler bandwidths. These two parameters characterise the performance of the adaptive notch filter in terms of noise rejection and signal tracking capabilities. Two criteria for setting the adaptation step of the PG algorithm adopted by the adaptive notch filter were also provided as a function of the loop and Doppler bandwidths, respectively. Algorithm extensions were also suggested and explained using results from the FLL theory. The theory developed is supported by Monte Carlo simulations and provides valuable insight on the behaviour of the adaptive notch filter and its extensions. In this Appendix, the proof of (51) is briefly outlined.
The two variances in the second line of (75) are equal due to the circular symmetry of η (1 − k a ) 2 k 2m a = s 4 4 The covariance term in (75)  12 Appendix 2: Proof of (53) In Section 5.2, it has been shown that C[n] is obtained by scaling and filtering a zero mean white Gaussian process with variance σ 2 /2. The filter used to generate C[n]i sd e fined by (49). In the equivalent model depicted in Fig. 4, N d [n] is processed by a filter with a transfer function which is a scaled version of (49). Thus, the noise term entering the loop in Fig. 4 has the same spectral properties of C [n]. In order to assign the same power of C[n] to the coloured noise term, the following condition has to be fulfilled where 4A 2 is the square of the scaling factor present in (48) which defines C[n]. Variance (53) directly follows from (80).