This paper describes how the fringe-washing function of a wide field of view interferometric radiometer can be accurately estimated using a relatively simple method. Fringe washing is an undesired effect that can be avoided in radio-telescopes due to its narrow-beam nature, but which is unavoidable in wide field of view interferometers, as those built for Earth Observation applications. The 3-delay method is based on performing correlations at -T, 0 and +T lags of each baseline of the interferometer, when all receiver inputs are connected to a common noise source (centralized correlated noise injection). In addition, a fringe-washing function closure relationship is presented which allows to extend this concept to aperture synthesis radiometers using distributed correlated noise injection. The results presented in this paper are simulations based on real measurements of the end-to-end frequency response of the first prototype receivers of ESA's Microwave Imaging Radiometer with Aperture Synthesis (MIRAS).
 Aperture synthesis microwave radiometers for Earth Observation are based on the concept of interferometry. A large synthetic aperture radiometer with a wide beam is prone to have appreciable fringe-washing effects because the signal coming from a slant direction θ will arrive with substantial delay between the two end receivers of the longest baseline D. Severe fringe-washing happens when the delay of the projected baseline is comparable to the correlation time of the signal (the inverse of its bandwidth B). Therefore, to neglect the fringe-washing, the following must happen (c being the speed of light)
Let us consider MIRAS, a real sensor under mission study at ESA, as an example. MIRAS is an L-band (1400-1427 MHz) Microwave Imaging Radiometer with Aperture Synthesis, to be flown as single payload on board the European Soil Moisture and Ocean Salinity (SMOS) mission (Figure 1).
 MIRAS has a very wide field of view of about 64° and consists of an array of 72 small receivers, arranged along three coplanar arms at 120° from each other (Figure 2). The spacing between elements is 0.88λ which gives an arm length of ∼ 4m, capable of a 2° − 2.5° angular resolution.
 The slant delay at the edge of the field of view (θ = 32°) corresponding to the maximum baseline of MIRAS, , is given by left side of equation 1:
and this is one fifth of the inverse of the noise equivalent bandwidth:
The condition of equation 1 is just barely fulfilled and thus, the fringe-washing effect has to be accounted for in MIRAS.
 It could be argued that an interferometer can be designed in a way that the fringe-washing is negligible. However, in practice, the cost of this is very high. In MIRAS it would require to partition the full 20 MHz band into narrower subbands for which the correlation time is long enough to neglect fringe-washing. But this leads to a number of correlators extremely high, as it increases times the number of subchannels. As a consequence, also the receiving chain and the calibration hardware complexity increases, contributing to increase of mass and volume of the satellite. In conclusion, fringe-washing cannot be avoided in wide field of view interferometry even with state of the art technology and has to be accounted and corrected for.
where is the Boltzmann constant, G0 the antenna gain, B the receiver noise equivalent bandwidth, α is the maximum modulus of the voltage gain of the receiver, F the normalized antenna voltage pattern, TB the polarimetric brightness temperature, f0 the central frequency, (ξ,η) the director cosines of a particular direction, (u, v) the baseline coordinates normalized to the central wavelength (c/f0), i and j the particular receivers and (τ) the fringe-washing function.
 The fringe-washing function is defined by
where f0 is the nominal center frequency of the RF band, Hn(f) is the frequency response of each receiver normalized to its maximum amplitude α and B is the noise equivalent bandwidth:
The fringe-washing function in equation 3 is complex in general. If the filter response is symmetric with respect to the center frequency f0 then the fringe-washing function is real.
 Let us assume a rectangular frequency response, the same for all receivers,
The amplitude of the fringe-washing function at the edge of the swath and for the longest baseline of MIRAS is, using the example above,
that is, about 10% amplitude attenuation. This has to be corrected for as MIRAS performances requires amplitude errors to be below 0.5%. In Camps et al.  a method for on-board determination of the fringe-washing function is explained for the first time. Our present paper provides a more simplified method and extends it to radiometers with distributed noise injection. We refer to this method as ”3-delay method” and it is described next.
3. Three-Delay Method
 The block diagram of every single channel receiver of an aperture synthesis radiometer is shown in Figure 3. The antenna has two probes, one for each polarization, and only one is selected at any time by an input switch. All the filtering and amplification provided by the electronics are represented by a single block with transfer function H(f). It is further assumed that the receiver performs a frequency down conversion by a complex local oscillator as depicted on Figure 3. Every receiver provides the in-phase and quadrature components of the input signal, which are then 1-bit digitized and cross-correlated for all possible pairs of receivers to obtain the visibility function above.
 The input switch has two other inputs for calibration signals: a u-input which is connected to a matched load and provides uncorrelated noise and a c-input which allows to inject correlated noise n(t) generated by a noise diode simultaneously into several receivers.
 Let us consider a specific pair of receivers during correlated noise injection. The complex cross-correlation between the analogue outputs of two receivers is
where Rhihj(τ) is the cross-correlation between the frequency responses
and Γninj, the cross-correlation of the injected noise at the input of the receivers [Panaro, 2001] (the receiver noise is uncorrelated across receivers and does not contribute)
δa(τ) being the analytic delta function and TN the equivalent temperature of the injected noise.
The cross-correlation of the 1-bit digitized version of the receiver outputs is a function of the normalized cross-correlation of the corresponding analogue signals [Thompson et al., 1986]
where the p superscript is used here to denote real or imaginary part, Γs(τ) is the autocorrelation function of the analogue signals and
with TR being the receiver noise equivalent temperature. Taking equations 13 and 15 into 14 yields
Equation 16 shows that the fringe-washing function can be determined provided (a) the noise injected into the receivers TN and their noise equivalent temperature TR are known, and (b) the cross-correlation of the output is measured for a reasonable set of values of the delay τ.
 The measurement of the cross-correlation at many delays requires costly hardware, a problem to be solved by the method proposed, which is able to accurately determine the fringe-washing function of a baseline by measuring the output cross-correlation at only 3 delays. The 3-delay method relies on the fact that the frequency responses of all receivers are fairly close to a nominal one. This fact has been demonstrated through an ESA MIRAS demonstrator breadboard development (frequency responses within ±0.5 dB in the passband) and can be assumed to be the real situation.
 A simple case is that of rectangular frequency responses (equation 5). Real filters are close to rectangular because high order filters are required to have a large out of band rejection. Assuming all filter blocks identical, the corresponding ideal fringe-washing function results in the well-known sinc function of equation 6, but the actual receiver frequency response is not perfectly rectangular in fact. The filters from different receives of the interferometer are not exactly equal to each other, exhibiting a spread in the cutoff frequencies, noise equivalent bandwidth, bandpass shape and group delay. This means that the fringe-washing function is not a pure sinc function, and varies from one baseline to another. However, as long as all filters are close to the nominal square filter, the fringe-washing functions of all baselines will also be close to the nominal sinc function.
 By measuring the cross-correlations at , and lags a sinc can be fit in amplitude as a good approximation to the actual fringe-washing function. Regarding the phase, the nominal rectangular filters have a linear phase. In practice, to better match the slightly deformed real frequency responses and to get full advantage of the three delays, a second order polynomial is used for phase fitting.
4. Simulation Results of the Three-Delay Method
 Four real MIRAS channel responses have been measured. Figure 4 shows the frequency responses of the channels and their group delays (estimated as ). The behavior of the filters, as far as the phase response is concerned, is not very linear, giving rise to some group delay mismatches between pairs of filters. This produces a shift of the peak of the fringe-washing function out of the origin as shown in Figure 5. Obviously this shift of the overall fringe-washing function cannot be neglected. On one hand the sensitivity at is decreased. On the other hand, the retrieval of the brightness temperature over the field of view (12.5 ns) is degraded because there is an amplitude decay and moreover the effect is asymmetric as the peak is not centered at the nominal position. In fact, it follows from (equation 2) that the fringe-washing term is spatially dependent and must be calibrated to the same error as the antenna voltage patterns.
 From zoomed Figure 5 one can see that the peaks appear shifted but still the shapes of the particular functions around these peaks might resemble the main lobe of the sinc function. The usual way MIRAS works is measuring the cross-correlation of the baselines for only and subsequently recovering the brightness temperature from the visibility function by solving the integral equation (2) for the unknown (ξ, η). In order to account for the shift of the fringe washing function a method which makes measurements of the cross-correlation at different time delays will be used. The aim is to fit two functions to the real modulus and to the phase of the fringe washing function around its maximum and over a range of delays corresponding to the alias free field of view of MIRAS (<12.5nsec). The fitted functions can then be used in an inversion algorithm based on numerical inversion of the integral equation (2) to recover the brightness temperature.
 As explained in previous section, high order filters which resemble a rectangular window function can be approximated within the alias free field of view by
measured in three points . For technical convenience, MIRAS uses , which is the sampling period of its channels. Unlike the A,B,C coefficients, which can be found numerically, the D,E,F require solving a linear system, so they can be explicitely given as
where Φ is the measured phase arg ij at the given delay.
 The real cross-correlation measurements must be regarded as noisy due to the finite integration time of the correlators. This adds an uncertainty to the previous approximations. Due to nonlinearity of the amplitude approximation, this uncertainty was assessed by statistical averages of realizations: noise was added to the three measured points ij(−T), ij(0) and ij(+T), then the approximations performed and the statistics calculated from the results.
 To relate the noise on the measured points to the integration time of the correlator the results presented by Camps et al. [1998a] were used. With the nomenclature introduced in Section 3 the variances on the real (p) and imaginary (q) parts of Γsisj (equation 13) are
where B is the bandwidth of the receiver channels, teff the effective integration time to account for the digital correlators of MIRAS, and the function Λ (Λ(x) = 1 − ‖x‖ for ‖x‖ < 1 and 0 elsewhere), is due to rectangular shape of the filters for which the result was derived. The Δf term is the offset of the oscillator frequency from the channel center (DSB or SSB receiver). With the probability density function being approximated by the Gaussian distribution (case of high Γs and high SNR during calibration), the variance on any of ij(−T), ij(0) or ij(+T) is
where and . For an 1-bit digital correlator at Nyquist sampling . MIRAS applies oversampling , then the coefficient of 2.46 drops to ≈2.1 [Camps et al., 1998b]. And finally for the SSB MIRAS receiver: ‖Δf‖ > B/2, hence .
Figure 4 showed four channel responses, say k,l,m,n of MIRAS. Then the fringe washing functions of the pairs kl,kn,mn,ln (Figure 5) were approximated. As an example, Figure 6 shows the approximation error for the kn pair, using the integration time , and the . The deviation due to approximation, as well as the uncertainity due to noise increase rapidly for ‖τ‖ > T. Of interest are the values within the field of view τ = ±12.5nsec, e.g. the two local maxima between the sample points. Their deviations were found to be below 10−3. In the case of the phase approximation the maximal deviations were 0.02°…0.03°.
5. Fringe-Washing Closure Relation
 This section shows a closure relationship for the fringe-washing function which makes it possible to apply the 3-delay method to an aperture synthesis radiometer with distributed noise injection [Torres et al., 1996]. Distributed noise injection consists of injecting correlated noise to groups of receivers instead to all receivers, as in a centralized configuration. The noise injection is done in two steps in such a way that the groups of receivers connected to a common noise source in the two steps partially overlap.
 More generally, a set of receivers k, l, m and n used in an aperture synthesis radiometer is assumed. Two groups of receivers, k − l and m − n, can be considered, each group connected to a different noise source during the first step of the distributed noise injection. Then using equation 16 the fringe-washing functions corresponding to the baselines and can be found.
 During the second step, the group l − m is formed by connecting the corresponding receivers to a different noise source, and the fringe-washing function is obtained.
 The closure relationship of the fringe-washing function is based on the Wiener-Khintchine theorem. This theorem states that the cross-correlation and the power cross-spectral density of two complex valued functions are related to each other by a Fourier Transform. Therefore, the Fourier Transform of the fringe-washing function is proportional to the cross power spectral density of the frequency responses of the receivers, translated by the center frequency. Considering the four receivers above and the fringe-washing functions obtained during steps 1 and 2, the following relationships hold
And similar relationships can be written for the m − n and l − m pairs. The power cross-spectral density functions can be combined in the following manner
that is, the fringe-washing function of the pair kn can be obtained from that of pairs kl, mn and lm even if the receivers k and n have never been connected to a common noise source.
 The fringe-washing closure expressed in equation 25 allows to extend the 3-delay method to aperture synthesis radiometers implementing a distributed noise injection rather than centralized noise injection. This is demonstrated for the case of MIRAS in the next section.
6. Simulation Results of the Fringe-Washing Closure
 Due to its considerable physical dimensions, MIRAS is applying distributed calibration approach [Torres et al., 1996]. In this scheme, rather than all receivers, always only a group of twelve receivers is fed with the calibration signal. The groups are chosen so that six receivers overlap, thus by two measurements the continuity of the phase along the arms of MIRAS can be assured. This scheme leaves many baselines being never driven by the same calibration signal and so the three delay method cannot be applied to them directly and the closure relationship, described in the previous section has to be applied instead.
 Taking the example of MIRAS (Figure 2): the hub and each of the three arm segments contain an independent noise source. In the 'odd' period, the noise source of the hub and the second arm supplies signal to the receivers: six in the hub and twelve in first two segments of each arm (receivers of last segment are not driven). Then in the second phase, each even noise source supplies the signal: the source in the first arm segment drives the hub and the first segment, and the source in the last segment drives the receivers in the second and third arm segments.
 As explained in Section 3, measuring the cross-correlations (τ) of the i-th and the j-th channel at three delays, the fringe washing function (τ) between those channels can be satisfactorily approximated. All cross-correlations (τ) measured in one integration can be collected into a matrix CΓ of size 72 × 72, as there are 72 receivers in all instrument. One can construct and matrices for the odd and even phases. Due to its considerable size, the CΓ matrix is schematically presented in Figure 7. The receivers were indexed in the following way: each submatrix of 24 × 24 represents receivers along one axis of MIRAS including the six receivers on the HUB. First six of each group of 24 index the HUB receivers, second six the first arm, etc.
 Most of the values in the CΓ matrices are zero as they are results of correlating signals which originate from independent sources. No fringe-washing function approximation can be performed for those. In Figure 7a the nonzero cross-correlations of are marked by light-grey squares and the nonzero Γdd's of are marked dark-grey. For these pairs the fringe-washing function approximation is possible.
 Applying the closure relationship (equation 25) on the approximated fringe-washing functions, all the other functions lying within the three 24 × 24 submatrices along the main diagonal (and also those up to the first arm segment), can be recovered. In Figure 7a, a rectangular line connects the four fringe-washing functions related by the closure relationship. If three of them are measured during the odd and even phases, the fourth can be calculated.
 Then, considering all these fringe-washing functions as known and applying the closure relationship the second time, the rest of the functions can be calculated. This is sketched in Figure 7b, where the line represents the second closure. It is noted that second closures include fringe-washing functions obtained in the first closure. This way, by two measurements (odd and even phase) and then applying the closure relationship twice all fringe-washing functions can be recovered.
 As an example, the which was described in the last paragraph of Section 4 and is shown in Figure 6, was recalculated by the closure relationship. The modulus and phase of in Figure 8 were estimated from the approximations of , and from the Section 4. As a consequence the deviations are nonzero at .
 Additionaly, to verify the case when the closure relationship has to be applied twice, we estimated also the , and functions by the closure relationship and used these estimates to calculate by applying the closure the second time. In this case, the standard deviations are further increased (Figure 9) when compared to previous results presented in Figures 6 and 8.
 The results presented in Figures 6, 8 and 9 are estimates from simulations as described in Section 4. The Fourier transform was approximated by DFT and a rectangular window of width was applied to data before the transformation (B being the nominal bandwidth).
 This article has presented a method for calibration of the fringe-washing function which is accurate enough to satisfy the requirements for the MIRAS: ∼ 0.5% in amplitude and 0.5° in phase [Camps et al., 1999]. It shows, that measurements on three delays are enough to approximate the fringe-washing function of a baseline by the sinc function in amplitude and a second order polynom in phase, if the channel responses are close to rectangular.
 Further, the article shows how to overcome a difficulty imposed by the distributed calibration method used in MIRAS, which makes the direct measurements of cross-correlations needed for the 3-delay method impossible. Section 5 presents the closure relationship which allows to calculate the fringe-washing function of a baseline even if it was never driven by a common source during the calibration. Applying the closure relationship twice all fringe-washing functions of the instrument can be recovered.
 These theoretical results were quantified on the example of two complex channels of the actual MIRAS instrument. Monte Carlo simulations were performed to asses the accuracy and sensitivity of the method. It was found that the worst case of the approximation errors within the field of view (‖τ‖ < 12.5nsec) were ∼0.1% in amplitude and ∼ 0.035° in phase, which is well below the requirements. For the fringe-washing functions recovered by the closure relation some degradation of precision is apparent (Figure 8), but even with the closure applied twice (Figure 9) the error is still below the required limits.
 The authors would like to thank MIER Comunicaciones, Barcelona (Spain) for providing the channel response data of the MIRAS receivers on which this article is based.