You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Radio interferometric observations are less susceptible to radio frequency interference (RFI) than single dish observations. This is primarily due to : (1) fringe-frequency averaging at the correlator output and (2) bandwidth decorrelation of broadband RFI. Here, we propose a new technique to improve RFI suppression of interferometers by replacing the fringe-frequency averaging process with a different filtering process. In the digital implementation of the correlator, such a filter should have cutoff frequencies $< 10^{-6}$ times the frequency at which the baseband signals are sampled. We show that filters with such cutoff frequencies and attenuation $>$ 40 dB at frequencies above the cutoff frequency can be realized using multirate filtering techniques. Simulation of a two element interferometer system with correlator using multirate filters shows that the RFI suppression at the output of the correlator can be improved by 40 dB or more compared to correlators using a simple averaging process.'
author:
- 'D. Anish Roshi'
- 'R. A. Perley'
title: A New Technique to improve RFI suppression in Radio Interferometers
---

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\#1 1.25in .125in .25in

Introduction
============

Radio interferometers provide high-resolution, high-dynamic range radio images of celestial objects. It is well known that radio interferometers are less susceptible to radio frequency interference (RFI) compared to single dish. However, the dynamic range of the images in many cases is limited by interference indicating the need to improve RFI suppression.

In this paper, we describe a new technique to improve RFI suppression in interferometers [^1] . The basic idea is to replace the averaging done at the output of the correlator with a multirate filtering process. A brief discussion of RFI suppression in existing interferometers is given in Sec. 2. Sec. 3 describes the proposed technique, gives the design of the multirate filter and presents simulation results. The limitations of the proposed technique are discussed in Sec. 4 and the effects of the multirate filtering process in the visibility and spatial domain are discussed in Sec. 5.

RFI suppression in existing Radio Interferometers {#secRFIim}
=================================================

Correlated and Uncorrelated RFI
-------------------------------

In the case of interferometric observations interference can be broadly classified as correlated and uncorrelated RFI. Consider a two element interferometer near a RFI source (Fig. 1). The voltages $V_1$ and $V_2$ at the outputs of the two antennas each consists of three components — the system noise ($V_{1sys}$, $V_{2sys}$), the source noise[^2] ($V_{1source}$, $V_{2source}$) and the RFI component ($V_{1rfi}$, $V_{2rfi}$). For the moment we will not consider fringe stopping, which is the case when the source is at the north pole. Also we consider that the RFI is not moving with respect to the interferometer. Interferometers usually use complex correlators, which are constituted of ‘real’ and ‘imaginary’ parts. For the discussion in this paper, we consider a ‘real’ correlator. But the analysis is equally applicable to the ‘imaginary’ part of the correlator as well. The mean value of the correlator output, $V_c$, is given by $$V_c = <V_1 V_2> = <V_{1source}V_{2source}> + <V_{1rfi}V_{2rfi}>,$$ where $<>$ denotes time average. If the separation between antennas (baseline length) is $>$ a few 100 kms, which is the case for the VLBI (very long baseline interferometry), and if the RFI source is near one of the antennas, then the second antenna will not pick up the RFI (ie say $V_{1rfi} = $ 0). This means that $V_c$ does not have any contribution from the RFI source or that the RFI is uncorrelated. If this condition is not true then the RFI is correlated (for this discussion we consider that the bandwidth of the system is very small so that there is no bandwidth decorrelation). In most connected interferometers RFI is correlated in many baselines. For the VLBI, most terrestrial RFI is uncorrelated, but satellite interference can be correlated in most ‘short’ baselines. The technique discussed in this paper to suppress interference deals with correlated RFI. Note that the RMS (root mean square) value of the correlator output is proportional to the geometric mean of the total powers at the inputs of the correlator if the source noise power is much smaller than the system noise power. The correlator input powers have contributions from the RFI even if it is uncorrelated. Thus uncorrelated RFI also limits the dynamic range of the image.

Fringe stopping and correlated RFI
----------------------------------

For simplicity consider that the baseline of the two element interferometer is in the East-West direction. The output of the two antennas are connected to a correlator with cables of equal length (Fig. 2). The response of the interferometer $P(\theta)$ is given by (Thompson, Moran, Swenson, 2001) $$P(\theta) = P_{ant}(\theta)\ \mbox{cos} \left(\frac{2\pi D \ \mbox{sin}(\theta)}{\lambda} \right),$$ where $P_{ant}(\theta)$ is the cross response of the two antennas, $D$ is the baseline length and $\lambda$ is the operating wavelength. $\theta$ is measured with respect to the zenith, which we equate to the hour angle ($HA$) in radians for simplicity. A source moving through the interferometer response produces “fringes” at the output of the correlator. The fringe frequency $f_{fringe}$ in the present case is given by, $$f_{fringe} = \frac{D}{\lambda} \mbox{cos}(HA) \frac{\mbox{d}(HA)}{\mbox{d}t} \mbox{cos}(\delta),$$ where $\delta$ is the declination of the source and $\frac{\mbox{d}(HA)}{\mbox{d}t}$ is in units of radians per sec. The general equation for the fringe frequency is (Thompson 2001) $$f_{fringe} = \frac{\mbox{d}(HA)}{\mbox{d}t} \ u \ \mbox{cos}(\delta),$$ where $u$ is the projected baseline in the East-West direction in units of $\lambda$. The above equation shows that the fringe frequency due to a source depends on the projected baseline and earth’s rotation rate ($= \frac{\mbox{d}(HA)}{\mbox{d}t}$). Now consider a terrestrial, stationary, correlated RFI. Since the position of the RFI does not change with respect to the interferometer, there will be no fringe due to the interference.

For observations with the interferometer, the interferometer is ‘phased’ in the direction of the interested radio source through out the observations. This process of phasing the interferometer (which includes both delay compensation and fractional phase compensation) is called ‘fringe stopping’. Fringe stopping essentially makes the correlator output constant due to a (point) source at the phase center. Sources away from the phase center produce fringes with frequencies equal to the difference between $f_{fringe}$ of the source at the phase center and that of the source away from the phase center. Thus, in the above case, larger the difference between the $HA$ of a source and that at the phase center the larger will be its ‘differential’ fringe frequency. This also means that the correlated RFI now produces a fringe with a frequency equal to $f_{fringe}$ of the source at the phase center because of fringe stopping.

Fringe averaging and RFI suppression
------------------------------------

Usually in interferometric observations, one is interested in mapping sources within the primary beam of the antenna. Typically the differential fringe frequencies due to these sources is much smaller than $f_{fringe}$ of the source at the phase center. The maximum differential fringe rate is given by $$\Delta f_{fringe} \sim \frac{\mbox{d}(HA)}{\mbox{d}t} \frac{\Omega_{antenna}}{\Omega_{synth}} \sim  
                        \frac{\mbox{d}(HA)}{\mbox{d}t} \frac{D}{D_{antenna}},$$ where $\Omega_{antenna}$ and $D_{antenna}$ are the beam and diameter of each element of an interferometer, $\Omega_{synth} \sim \lambda/D$ is the synthesized beam, $\lambda$ is the operating wavelength and $D$ is the baseline length. Consider, for example, $D$ = 10 km and $D_{antenna}$ = 25 m. The differential fringe frequency is $\sim$ 0.005 Hz while $f_{fringe}$ of the source at the phase center is $\sim$ 3.5 Hz, for $\lambda = $ 0.21 m. The low differential fringe frequency allows one to average the output of the correlator for several seconds. Since the RFI produces a fringe (due to fringe stopping process) this averaging reduces the amplitude of the RFI at the correlator output thus making the interferometer less susceptible to RFI (see Perley 2002 for a quantitative estimate of RFI attenuation in Very Large Array due to fringe averaging). An additional suppression of broadband RFI occurs due to bandwidth decorrelation, which is a bonus over the former effect. We will not discuss the bandwidth decorrelation effect any further here (see Thompson 2001).

A new technique to improve RFI suppression in Interferometers {#sectech}
=============================================================

The averaging at the correlator output can be thought of as a moving average filtering process. The response of such a filter is shown is Fig. 4. The stop-band attenuation (i.e. the attenuation of frequencies beyond the cutoff frequency) of the filter determines the amount of RFI suppression achieved in this process if the fringe due to RFI is beyond the cutoff frequency of the filter. As seen from Fig. 4, stop-band attenuation in the worst case is 13.5 dB. We propose that RFI suppression in interferometers can be improved by using a filter with high stop-band attenuation at the output of the correlator. In the following sections we show how such filters can be designed and present results of the simulation done to tests its performance.

Multirate filtering
-------------------

Present day correlators are designed using digital techniques. The typical bandwidth of the interferometer is a few tens of MHz or more. Sampling the signals at Nyquist rate implies that the data rate at the output of the multiplier will be of the order of the bandwidth. As mentioned above the output of the correlator can be averaged for several secs. This means that the required cutoff frequency of the filter that replaces the simple averaging should be a few tenths of a Hz. Thus the normalized (normalized with half the Nyquist frequency of the baseband) cutoff frequency of the filter is $< 10^{-6}$. A direct realization of a digital filter is not practical in such situations. However, such filters are realizable using multirate filtering technique. This technique essentially implements the required filter after reducing the sampling frequency (see Fig. 3; Bellanger 2000).

The basic filter used for the multirate filtering design is a digital elliptic filter. Using Matlab, a digital elliptic filter of order 10 with normalized cutoff frequency of 0.1 and stopband attenuation of 100 dB is designed. The passband ripple is chosen as 0.1 dB. The elliptic filter is realized using IIR (infinite impulse response) filter sections and its response is shown in Fig. 3. The sampling rate reduction is done in two ways: (1) using polyphase filtering technique; (2) by low-pass filtering the data using a moving average filter and then resampling the data at half the number of points averaged. The second type of filter is economical and simpler to implement in hardware. For testing the performance of the filter the sampling rate is reduced by 100. Thus the effective cutoff frequency of the filter is $10^{-3}$ and we believe that this could be scaled further down by appropriately resampling the data. The responses of the multirate filters are shown in Fig. 4.

Results of the simulation
-------------------------

A simulation was done in Matlab to determine the improvement in RFI rejection of a two element interferometer by implementing the multirate filtering process. Fig. 5 shows the block diagram of the interferometer implemented for the simulation. The fringe generator adds a fringe to the source noise. The antenna blocks are power combiners, combining the source noise, the two independent system noises and the RFI. All noises are Gaussian random variables. The response of the receiver is realized using the receiver filters. Fringe stopping is done before multiplying the two antenna outputs. The relative delay between the two antennas is made Zero for the simulation.

The designed multirate filters and the moving average filter are used as post-detection filters to study their relative performance. A sine wave with normalized frequency of 0.65 is used as RFI. The receiver bandwidth in normalized frequency is selected as 0.2 centered at 0.625. The receiver noise ($T_{sys}$) and source noise powers are made equal to 0.2 in arbitrary units. Fig. 6 shows the spectra of the two receiver outputs. The source noise is multiplied with a fringe of normalized frequency $3 \times 10^{-3}$. If we consider that this fringe freq should translate to say 3 Hz, then the Nyquist frequency will be 2.0 KHz. This choice of the fringe frequency is limited by the practicality of the simulation. The outputs of the correlator for different types of post-detection filters are shown in Fig. 7. The remarkable ($>$ 40 dB) improvement in the RFI suppression compared to the case when a moving average filter is used as post-detection filter is evident from this figure. Comparing the spectrum of the correlator output with a moving average post-detection filter and multirate post-detection filter shows that the spectral values below the cutoff freq of the filter are almost the same for both cases. This means that the amplitudes of the desired fringe frequencies are not affected by the multirate filtering process.

The effectiveness of the technique is also determined by estimating the spectral power of the RFI at the output of the correlator with different types of post-correlation filters. To estimate the spectral power of RFI a power spectrum of the correlator output is first made. The difference of the spectral values of the channels where the RFI is expected and the average of the values of the two nearby channels is then taken. If the RFI power is comparable to the noise fluctuations in the spectrum then the difference can also be negative. We assigned a value of zero for the RFI spectral power if the estimate is negative. Fig. 6 shows the spectral power of RFI vs the interference-to-noise ratio (INR) of the RFI at the output of the receiver. The INR is the ratio of the RFI power to the noise power (receiver noise + source noise) at the output of the receiver. The remarkable suppression of the spectral power of the RFI with a multirate filter in post-detection stage is clear from Fig. 6.

Limitations of the technique {#seclim}
============================

The technique discussed in this paper to improve RFI suppression of interferometers is effective only for correlated RFI (see Sec. 2). Usually terrestrial RFI is correlated on baselines of length less than a few 100 kms. Correlated RFI produces fringes at the output of the interferometer. The fringe frequency depends on the baseline length as well as the source declination for a given hour angle. Effective suppression of RFI occurs if the fringe frequency is much above the cut-off frequency of the multirate post-detection filter (see Sec. 3). However, if the baselines are small ($<$ a few kms) then the fringe frequency will be small and it becomes difficult to make filters that can attenuate these frequencies. Thus the technique presented here gives good RFI rejection for intermediate ($\sim$ a few kms to a few 100 kms) baselines. The fringe frequency also becomes smaller for sources near the north celestial pole (declinations near 90), which means that this technique will not work efficiently for this case as well.

Often RFI will have a strong carrier and associated modulating frequency components. The modulation results in a wider spectrum of the RFI at the output of the correlator. Thus only those components below the cut-off frequency of the multirate filter will be attenuated.

The analysis and simulation presented in this paper consider that the RFI is not time variable. However, in reality RFI is time variable. The response of the multirate filter to transient RFI needs to be studied. This will be presented elsewhere.

Effects of multirate filtering on the visibility and spatial domain {#secvis}
===================================================================

Till now the effect of the multirate filtering technique is discussed in relation with the time series at the output of the correlator. Any processing of the correlator output has an equivalent effect on the visibility domain and correspondingly on the spatial domain (i.e. positions in sky). The effective position of a stationary RFI is the north celestial pole. Thus suppressing the RFI using the multirate filtering process is equivalent to introducing a spatial null at the north celestial pole. In fact, the spatial null is not confined to the north celestial pole alone but extends along the spatial domain where the differential fringe frequencies are larger than the cutoff frequency of the multirate filter. The attenuation due to the spatial null will be proportional to the attenuation of the RFI due to the multirate filter.

In the visibility domain, the multirate filtering process essentially convolves the true visibility with the filter response along $uv$ tracks traced by pairs of antennas in an interferometer. Depending on the filter response, the amplitude and phase of the spatial frequencies of interest are modified. Since the filter is implemented digitally its response is known well and is identical for all the correlators. With the knowledge of this filter response, we believe a correction similar to the primary beam correction can be done to take out the effect of filtering. A detailed discussion on this will be presented elsewhere.

Acknowledgment
==============

DAR thanks <PRESIDIO_ANONYMIZED_PERSON> for all useful discussions during the course of this work. The basic idea described in this paper stemmed from a discussion DAR had with Prof. A. Pramesh Rao, National Center For Radio Astrophysics, Pune, India.

Bellanger, M., 2000, Digital Processing of Signals, John Wiley & Sons, Ltd, Chichester, UK.

Perley, R., 2002, EVLA Memo No. 49, NRAO Library, Socorro.

Sachdev, S., Udayashankar, N., 1995, BASI, 23, 574.

Thompson, A. R., Moran, J. M., Swenson, Jr., G. W., 2001, Interferometry and Synthesis Technique, John Wiley & sons, inc., New York.

[^1]: To our knowledge a technique similar to that discussed in this paper was used earlier in Mauritius Radio Telescope data reduction (Sachdev, Udayashankar 1995).

[^2]: The source noise is that component of the sky noise which is not ‘resolved’.