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 Formation of the null at the phased-array beam pattern pointed toward the source of a radio frequency interference (RFI) can serve to mitigate the effects of that RFI. Several algorithms for creating a null at the phased-array beam pattern have been designed. On the basis of an array side lobe minimization algorithm (Kogan, 2000), a new method for creating a null at the phased-array beam pattern is described here. The algorithm is a straightforward one and can have an advantage in creating wide-area nulls, especially for phased arrays with a small number of antennas. We have tested this algorithm for a possible Square Kilometre Array antenna station configured as a phased array of 19 antennas and for the Very Large Array configuration D (VLA-D), working as a phased array for very long baseline interferometry experiments. The null is created at the given area around the given direction. The narrow null (∼0.1 ) deeper than 100 dB level can be achieved at the cost of a 2% decrease in the signal-to-noise ratio in the main direction. The creation of such a deep null can be impractical for many reasons, however, first and foremost because the phases and weights would have to be specified with impractical precision. Rounding the phase and weights to 1° and 0.01, respectively raises the null up to ∼50 dB.
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 The idea of null formation at a phased-array beam pattern to mitigate the effects of radio frequency interference (RFI) has been widely discussed in the literature. Null formation in given directions has been discussed by Godara , Ellingson and Hampson , Smolders and Hampson , Ellingson and Cazemier  and others. Bower  applied the idea to simulate null formation for data from the Allen Telescope Array (ATA) which have 350 antennas. He showed possibility of very deep (deeper than 100 dB) but also very narrow nulls. Harp  designed an iterative method for generating wider area nulls at the beam pattern of a phased array. He applied his method to ATA and obtained approximately the same depth of the narrow nulls as Bower . His algorithm can provide the wider nulls at the cost of a visible loss in the SNR. Another algorithm for creating a null at the phased-array beam pattern is described here. This algorithm is a modification of the algorithm designed to minimize the side lobes of an array [Kogan, 2000]. This method is a straightforward one, and can have an advantage in creating a wide-area null, especially for a phased array with a small number of antennas.
2. Description of the Algorithm
 The beam power pattern of a phased array can be described by the following expressions:
unit vector directed toward the point in the sky;
unit vector pointed toward the target source;
vector describing the position of antenna n;
number of antennas in the array;
desired weight and phase to add to antenna n to create the null near .
 It is assumed in equation (1) that the array is ideally phased in the 0 direction. So the value of B() in the direction of the target source ( = 0) depends only on the added weights and phases wn, ϕn. The positions of the antennas n are normalized to the maximum baseline D. So ∣n∣ is nondimensional and ∣n∣ ≤ 0.5. The components of are measured in .
 The derivatives of B in respect to ϕk and wk are given by the following expressions:
 If we want to minimize the value of B in the direction, we need to modify ϕk and wk proportionally with their relevant derivatives (gradient descent). This is the method utilized for the optimization of an array configuration to minimize side lobes [Kogan, 2000]. Now we apply the same idea to find the correction in the phase and weight of each antenna to create the null, in the vicinity of a given direction in the sky. The creation of this null is an iterative process. At each iteration, the algorithm modifies the phase and/or weight of each antenna in the array by taking a small step size in the negative gradient direction (found through the derivatives given above). The value of this step size is proportional to the relevant derivative and is controlled by a user separately for weight and phase. A larger step size might improve convergence speed, but may also cause the algorithm to fail.
 The algorithm is implemented as an AIPS [Greisen, 2003] task (private version of AIPS). It is clear that creating the nulls by adding phase and weight to each antenna decreases the signal-to-noise ratio, observing the target source through the main lobe of the array (0 direction). The reduction of the signal-to-noise ratio (SNR) can be calculated by the following expression:
where the numerator describes the power of the collected signal, reduced in comparison to the case ϕk = 0, wk = 1; the denominator describes the power of the noise in the collected signal; and the antennas in the array are assumed identical. If all antennas are weighted equally without a phase shift (i.e., wk = 1 and ϕk = 0 for all k) then SNR = 1. Otherwise SNR < 1.
 The position of the null and its width are measured (in this algorithm) by the number of the synthesized beam width , where D is the array size (maximum base line). So if the position of the null is determined as θ = α, then the width of the null on the frequency axis is determined by Δθ = α. Remembering that the space width of the null is determined as Δθ = Δα we infer that relative frequency bandwidth of the null and its relative space width are related by the following equation:
It is found empirically that the spatial width of the null measured in (Δα) does not depend on the null position measured in also (α). Therefore we can infer from equation (5) that a larger α (the null area located further from the main beam) corresponds to a lower bandwidth.
3. Square Kilometre Array (SKA) Antenna Station
 We tested this algorithm using a possible SKA antenna station [Kogan, 2003] configured as a phased array of 19 antennas, with the array size ∼100 m (Figure 1). The synthesized beam width of this antenna station is 400″ (λ = 20 cm, D = 100 m). We simulated a null located 10° away from the main lobe (α = 90). We easily got a null of depth 125 dB with a width of 65″ (Δα = 0.16). Unfortunately, the phases and weights had to be represented with at least 6 digit precision after the decimal point, what is impractical. We found that the depth of the null decreases dramatically when the implemented phase/weight correction is rounded off to 1° phase increments and 0.01 weight increments. Figure 2 shows the effect this rounding of weights and phase has on the beam pattern null located 10° away from the main lobe. This rounding causes the depth of the null to drop from 123 dB to 49 dB. For a null width of 200″ (Δα = 0.5) and a null location 10° away from the main lobe (α = 90), the bandwidth at the central frequency of 1440 MHz can be estimated using equation (5) as Δf = f · = 1440 · 0.5/90 = 8 MHz. Table 1 shows the parameters of the array beam pattern at the created null area.
Table 1. Parameters of the Null Located 10° Away From the Main Lobe (SKA Example)a
Null Depth, dB
Null Width, 30 dB, arc sec
Frequency = 1440 MHz; = 400″.
Phase/weight correction is rounded to 1° increments in the phase and 0.01 increments in weight.
4. Very Large Array Configuration D (VLA-D) Phased Array
 We tested the current algorithm for VLA-D, used as an array element for very long baseline interferometry experiments, when all 27 VLA antenna are phased together. The array size is ∼1000 m (Figure 3). The synthesized beam of the VLA-D as an antenna station is 40″ (λ = 20 cm, D = 1000 m). We simulated the null located 10° (α = 900) away from the main beam direction We achieved the null as deep as ∼130 dB with the width 8″ (Δα = 0.2). The bandwidth of the null at the central frequency of 1440 MHz can be estimated using equation (5) as Δf = f · = 1440 · 0.2/900 ∼ 0.3 MHz. Table 2 shows the parameters of the array beam patterns at the created null area including the rounding of the introduced weights and phases.
Table 2. Parameters of the Null Located 10° Away From the Main Lobe (VLA-D Example)a
Null Depth, dB
Null Width, 30 dB, arc sec
Frequency = 1440 MHz; = 40″.
Phase correction is rounded to 1°, and weight ≡ 1.
 The above-described algorithm is very straightforward in its computation of the weights and/or phases for each antenna of the phased array, to create the null at the given area of the array beam pattern. This algorithm is implemented as an AIPS task (the private version of AIPS). Using an SKA antenna station and the VLA-D array as examples, it is shown that the designed method allows creating nulls as deep as >120 dB to be obtained, while taking a sensitivity loss of <2%. Unfortunately this requires the phases and weights added to each antenna to be given with more than a six digit precision after the decimal point. This is clearly impractical and is a general problem in null formation. Rounding the phase/weight to 1° for phase and 0.01 for weight lifts the null depth up to ∼40–60 dB. The further the null is located from the main beam, the narrower the frequency bandwidth of the null will be. A difference in atmospheric delay for different antennas in the array can affect the depth of the null.
Thompson  noticed that since the interference is received through the far side lobes of the array antennas, deterministic nulling introduces requirements on the phase response of the far side lobes that are not usually encountered. Nulls deeper than 100 dB may not be required (for the array as an antenna station) because the RFI signal is considered at such a case ∼100 db stronger than the signal of the target source. It can lead to the individual antenna receiver saturation.
 I am grateful to the anonymous referees for their valuable comments which led to the improvement of the paper. I would like to thank Dima Kogan for his help with the English in this paper. The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.