Delay Calibration - Revision history

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:53:33Z

2, Cross-Checks

← Older revision		Revision as of 06:53, 20 March 2026
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	Note that any RF frequency above 2.15 GHz results in a negative IF frequency, which, when aliased about zero, becomes the absolute value of the IF frequency. We conclude that equation (3) is accurate for any ADC clock and RF frequency.		Note that any RF frequency above 2.15 GHz results in a negative IF frequency, which, when aliased about zero, becomes the absolute value of the IF frequency. We conclude that equation (3) is accurate for any ADC clock and RF frequency.

	Looking now at equation (2), the first term is constant over a given tuned band, while the second term is the channel-dependent phase associated with the residual fine delay <math>(\tau_g-\tau_1)</math>, i.e. the difference between the integer-stepped coarse delay and the true geometric delay. Let us rewrite equation (2) as		Looking now at equation (2), the first term is constant over a given tuned band, while the second term is the channel-dependent phase associated with the residual fine delay <math>(\tau_g-\tau_1)</math>, i.e. the difference between the integer-stepped coarse delay and the true geometric delay. Let us rewrite equation (2) as: <math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO})\tau_1 + \omega_{RF}(\tau_g-\tau_1).</math>

	:<math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO})\tau_1 + \omega_{RF}(\tau_g-\tau_1).</math>

	The first term, <math>(\omega_{ADC} + \omega_{VLO} - \omega_{FLO})\tau_1</math>, is constant for a given band, and by definition constant over a 1-s period since delay steps can only happen on 1-s boundaries. In fact, this term can remain constant for several minutes for short baselines that are not changing projected length very fast. The term can grow very large, and has stepwise discontinuities since <math>\tau_1</math> varies in coarse-delay steps. Because <math>\omega_{VLO}</math> ranges from 21.5-38 GHz, for the case of an 800 MHz ADC clock the frequency term ranges from 7.226 radians/ns to 110.898 radians/ns (414-6354 degrees/ns). Since the steps in <math>\tau_1</math> occur in 1.25 ns steps for an 800 MHz ADC clock, this is 517.5-7942.5 degrees/step. Although this seems large, it agrees with my analysis detailed in section 2.1 Delay Tracking, in the memo ''EOVSA_Calibration''. There I found a maximum fringe rate of 6.6 Hz. Taking 6354 degrees/ns, and maximum delay rate of 0.364 ns/s from that memo, I have (6354 degrees/ns)(0.364 ns/s)/(360 degrees) = 6.44 Hz. The slight discrepancy is due to the above numbers referring to a frequency of 17.650 GHz, whereas the memo used 18 GHz. Applying the correction, I get a fringe rate of 6.58 Hz.		The first term, <math>(\omega_{ADC} + \omega_{VLO} - \omega_{FLO})\tau_1</math>, is constant for a given band, and by definition constant over a 1-s period since delay steps can only happen on 1-s boundaries. In fact, this term can remain constant for several minutes for short baselines that are not changing projected length very fast. The term can grow very large, and has stepwise discontinuities since <math>\tau_1</math> varies in coarse-delay steps. Because <math>\omega_{VLO}</math> ranges from 21.5-38 GHz, for the case of an 800 MHz ADC clock the frequency term ranges from 7.226 radians/ns to 110.898 radians/ns (414-6354 degrees/ns). Since the steps in <math>\tau_1</math> occur in 1.25 ns steps for an 800 MHz ADC clock, this is 517.5-7942.5 degrees/step. Although this seems large, it agrees with my analysis detailed in section 2.1 Delay Tracking, in the memo ''EOVSA_Calibration''. There I found a maximum fringe rate of 6.6 Hz. Taking 6354 degrees/ns, and maximum delay rate of 0.364 ns/s from that memo, I have (6354 degrees/ns)(0.364 ns/s)/(360 degrees) = 6.44 Hz. The slight discrepancy is due to the above numbers referring to a frequency of 17.650 GHz, whereas the memo used 18 GHz. Applying the correction, I get a fringe rate of 6.58 Hz.

	The second term, <math>\omega_{RF}(\tau_g-\tau_1)</math>, can also be quite large. Since in general <math>(\tau_g-\tau_1)</math> can range from <math>-1/2</math> step (<math>-0.625</math> ns) to <math>+1/2</math> step (<math>+0.625</math> ns), for an RF frequency of 18 GHz, <math>\omega_{RF}(\tau_g-\tau_1)</math> ranges from -4050 to +4050 degrees. It is puzzling that the total range, 8100 degrees, is not quite the same as the max degrees/step of the natural fringe term (7942.5 degrees). However, the frequency 17.650 GHz does yield an exact match, which corresponds to 0 MHz in the band-34 IF, according to equation (3).		The second term, <math>\omega_{RF}(\tau_g-\tau_1)</math>, can also be quite large. Since in general <math>(\tau_g-\tau_1)</math> can range from <math>-1/2</math> step (<math>-0.625</math> ns) to <math>+1/2</math> step (<math>+0.625</math> ns), for an RF frequency of 18 GHz, <math>\omega_{RF}(\tau_g-\tau_1)</math> ranges from -4050 to +4050 degrees. It is puzzling that the total range, 8100 degrees, is not quite the same as the max degrees/step of the natural fringe term (7942.5 degrees). However, the frequency 17.650 GHz does yield an exact match, which corresponds to 0 MHz in the band-34 IF, according to equation (3).

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:52:23Z

2, Cross-Checks

← Older revision		Revision as of 06:52, 20 March 2026
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	is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo [[Downconversion and Frequency Tuning\|Documentation of Downconversion and Tuning]] (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).		is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo [[Downconversion and Frequency Tuning\|Documentation of Downconversion and Tuning]] (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).

	[[File:6.png\|~~500px~~\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]		[[File:6.png\|400px\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]

	Using equation (1), the variable LO would be tuned to 22.5 GHz for this band, so:		Using equation (1), the variable LO would be tuned to 22.5 GHz for this band, so:

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:52:12Z

2, Cross-Checks

← Older revision		Revision as of 06:52, 20 March 2026
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	is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo [[Downconversion and Frequency Tuning\|Documentation of Downconversion and Tuning]] (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).		is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo [[Downconversion and Frequency Tuning\|Documentation of Downconversion and Tuning]] (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).

	[[File:6.png\|~~600px~~\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]		[[File:6.png\|500px\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]

	Using equation (1), the variable LO would be tuned to 22.5 GHz for this band, so:		Using equation (1), the variable LO would be tuned to 22.5 GHz for this band, so:
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	Likewise, if the ADC clock frequency is 800 MHz, as it is at present, then the lower part of the band (0-200 MHz) is overlapped, and the upper part of the band (200-400 MHz) is direct relative to the RF (see Figure 3, reproduced from the earlier memo).		Likewise, if the ADC clock frequency is 800 MHz, as it is at present, then the lower part of the band (0-200 MHz) is overlapped, and the upper part of the band (200-400 MHz) is direct relative to the RF (see Figure 3, reproduced from the earlier memo).

	[[File:7.png\|thumb\|~~400px~~\|left\|Figure 3: Schematic representation of the third EOVSA downconversion by the digitizer, when the digitizer clock is at the non-ideal frequency of 800 MHz. The second IF band is shown in black (in MHz), while the mirrored IF band partially overlaps and extends to the left, marked in blue. The green block indicates the downconverted, digitized bandpass, whose scale is shown in blue (in MHz). The part of the band contaminated with overlapping is shown as the darker green hatched area.]]		[[File:7.png\|thumb\|350px\|left\|Figure 3: Schematic representation of the third EOVSA downconversion by the digitizer, when the digitizer clock is at the non-ideal frequency of 800 MHz. The second IF band is shown in black (in MHz), while the mirrored IF band partially overlaps and extends to the left, marked in blue. The green block indicates the downconverted, digitized bandpass, whose scale is shown in blue (in MHz). The part of the band contaminated with overlapping is shown as the darker green hatched area.]]

	In this case, for the same 2.0-2.5 GHz band 3 as the earlier example, we expect 2.0 GHz RF to be at 150 MHz IF, 2.15 GHz to be at 0 MHz IF, 2.3 GHz to be again at 150 MHz, and 2.5 GHz RF to be at 350 MHz. For these four cases, equation (3) gives:		In this case, for the same 2.0-2.5 GHz band 3 as the earlier example, we expect 2.0 GHz RF to be at 150 MHz IF, 2.15 GHz to be at 0 MHz IF, 2.3 GHz to be again at 150 MHz, and 2.5 GHz RF to be at 350 MHz. For these four cases, equation (3) gives:

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:50:51Z

2, Cross-Checks

← Older revision		Revision as of 06:50, 20 March 2026
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	:<math>f_{IF} = 1.2 - 2.5 + 22.5 - 21.15 = 0.05\ {\rm GHz} = 50\ {\rm MHz},</math>		:<math>f_{IF} = 1.2 - 2.5 + 22.5 - 21.15 = 0.05\ {\rm GHz} = 50\ {\rm MHz},</math>

	:<math>f_{IF} = 1.2 - 2.0 + 22.5 - 21.15 = 0.55\ {\rm GHz} = 550\ {\rm MHz}.</math>		:<math>f_{IF} = 1.2 - 2.0 + 22.5 - 21.15 = 0.55\ {\rm GHz} = 550\ {\rm MHz}.</math>

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:50:24Z

2, Cross-Checks

← Older revision		Revision as of 06:50, 20 March 2026
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	:<math>f_{IF} = 1.2 - 2.0 + 22.5 - 21.15 = 0.55\ {\rm GHz} = 550\ {\rm MHz}.</math>		:<math>f_{IF} = 1.2 - 2.0 + 22.5 - 21.15 = 0.55\ {\rm GHz} = 550\ {\rm MHz}.</math>

	Equation (3) works for any band, when <math>\omega_{~~VLO~~}</math> and <math>\omega_{~~ADC~~}</math> are changed appropriately.		Equation (3) works for any band, when <math>\omega_{RF}</math> and <math>\omega_{VLO}</math> are changed appropriately.

	Likewise, if the ADC clock frequency is 800 MHz, as it is at present, then the lower part of the band (0-200 MHz) is overlapped, and the upper part of the band (200-400 MHz) is direct relative to the RF (see Figure 3, reproduced from the earlier memo).		Likewise, if the ADC clock frequency is 800 MHz, as it is at present, then the lower part of the band (0-200 MHz) is overlapped, and the upper part of the band (200-400 MHz) is direct relative to the RF (see Figure 3, reproduced from the earlier memo).

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:44:37Z

2, Cross-Checks

← Older revision		Revision as of 06:44, 20 March 2026
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	:<math>\omega_{IF} = \omega_{ADC} - \omega_{RF} + \omega_{VLO} - \omega_{FLO} \qquad (3)</math>		:<math>\omega_{IF} = \omega_{ADC} - \omega_{RF} + \omega_{VLO} - \omega_{FLO} \qquad (3)</math>

	is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo ''Documentation of Downconversion and Tuning'' (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).		is the IF frequency corresponding to RF frequency <math>\omega_{RF}</math>. As shown in the memo [[Downconversion and Frequency Tuning\|Documentation of Downconversion and Tuning]] (see Figure 2, reproduced from that document), for an ADC clock frequency of 1200 MHz, we expect (blue numbers in Figure 2) an RF frequency of, say, 2.5 GHz, to be at IF frequency 50 MHz when tuned to the 2-2.5 GHz band (FSeqList = 3), while the other end of the band, 2.0 GHz, should be at 550 MHz (i.e. the IF band is inverted relative to the RF).

	[[File:6.png\|600px\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]		[[File:6.png\|600px\|right\|thumb\|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer. The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz). The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]

Sjyu: /* 2, Cross-Checks */

2026-03-20T06:41:49Z

2, Cross-Checks

@@ Line 106: / Line 106: @@
 === 2, Cross-Checks ===
 To verify that the above is correct, first note that the frequency
-[[File:6.png|600px|right|thumb|Figure 2: Schematic representation of the third EOVSA downconversion by the digitizer.  The filtered second IF band on the left, whose frequency scale is marked in black (in MHz), is mirrored and converted to the IF band on the right, marked in blue (in MHz).  The 600 MHz-wide digitized bandpass is shown in green, while the narrower 500 MHz target bandpass is shown by the inner dashed lines on the right.]]
+:<math>\omega_{IF} = \omega_{ADC} - \omega_{RF} + \omega_{VLO} - \omega_{FLO} \qquad (3)</math>
 Using equation (1), the variable LO would be tuned to 22.5 GHz for this band, so:
+:<math>f_{IF} = 1.2 - 2.5 + 22.5 - 21.15 = 0.05\ {\rm GHz} = 50\ {\rm MHz},</math>
-The equation (3) works for any band, when   and   are changed appropriately.
 Likewise, if the ADC clock frequency is 800 MHz, as it is at present, then the lower part of the band (0-200 MHz) is overlapped, and the upper part of the band (200-400 MHz) is direct relative to the RF (see Figure 3, reproduced from the earlier memo).
-[[File:7.png|thumb|400px|left|Figure 3: Schematic representation of the third EOVSA downconversion by the digitizer, when the digitizer clock is at the non-ideal frequency of 800 MHz.  The second IF band is shown in black (in MHz), while the mirrored IF band partially overlaps and extends to the left, marked in blue.  The green block indicates the downconverted, digitized bandpass, whose scale is shown in blue (in MHz).  The part of the band contaminated with overlapping is shown as the darker green hatched area.]]
+[[File:7.png|thumb|400px|left|Figure 3: Schematic representation of the third EOVSA downconversion by the digitizer, when the digitizer clock is at the non-ideal frequency of 800 MHz. The second IF band is shown in black (in MHz), while the mirrored IF band partially overlaps and extends to the left, marked in blue. The green block indicates the downconverted, digitized bandpass, whose scale is shown in blue (in MHz). The part of the band contaminated with overlapping is shown as the darker green hatched area.]]
-In this case, for the same 2.0-2.5 GHz band 3 as the earlier example, we expect 2.0 GHz RF to be at 150 MHz IF, 2.15 GHz to be at 0 MHz IF, 2.3 GHz to be again at 150 MHz, and 2.5 GHz RF to be at 350 MHz.  For these four cases, equation (2) gives:
+The second term, <math>\omega_{RF}(\tau_g-\tau_1)</math>, can also be quite large. Since in general <math>(\tau_g-\tau_1)</math> can range from <math>-1/2</math> step (<math>-0.625</math> ns) to <math>+1/2</math> step (<math>+0.625</math> ns), for an RF frequency of 18 GHz, <math>\omega_{RF}(\tau_g-\tau_1)</math> ranges from -4050 to +4050 degrees. It is puzzling that the total range, 8100 degrees, is not quite the same as the max degrees/step of the natural fringe term (7942.5 degrees). However, the frequency 17.650 GHz does yield an exact match, which corresponds to 0 MHz in the band-34 IF, according to equation (3).

Sjyu: /* 1. Background */

2026-03-20T06:40:08Z

1. Background

← Older revision		Revision as of 06:40, 20 March 2026
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	<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \tau_1 + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>		<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \tau_1 + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>

	Here <math>\tau_1</math> is the ~~non-~~integer "fine delay," which must be applied on a channel-by-channel basis across the 600 MHz IF band, while the first term is constant over the band for a given tuning frequency. The phase in equation (2) is to be subtracted from the measured baseline phase.		Here <math>\tau_1</math> is the integer "coarse delay," while the residual fine delay is <math>\tau_g-\tau_1</math>, which must be applied on a channel-by-channel basis across the 600 MHz IF band, while the first term is constant over the band for a given tuning frequency. The phase in equation (2) is to be subtracted from the measured baseline phase.

	=== 2, Cross-Checks ===		=== 2, Cross-Checks ===

Sjyu: /* 1. Background */

2026-03-20T06:35:53Z

1. Background

← Older revision		Revision as of 06:35, 20 March 2026
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	[[File:5.png\|right\|thumb\|600px\|Figure 1: Diagram showing the three downconversions.]]		[[File:5.png\|right\|thumb\|600px\|Figure 1: Diagram showing the three downconversions.]]

	As shown in Figure 1, a plane wave arrives later at the antenna on the left, so its fluctuating waveform is shifted by a phase proportional to the RF angular frequency times the geometric delay. Here the geometric delay <math>\tau_g</math> is the continuously varying delay required to track the source. The first downconversion by the variable LO inverts the frequencies on both antennas and shifts them by the variable-LO frequency. The second downconversion by the fixed LO inverts and shifts the frequencies by the fixed-LO frequency in a similar manner. This produces an IF frequency in the range 600-1200 MHz. The final downconversion is done by the ADC, which does a final inversion of the frequencies and shifts them by the ADC clock frequency. After digitization, an integer "coarse delay" <math>\~~tau~~</math>, rounded to the nearest digitizer clock step, is inserted into the right-hand antenna. After correlation (multiplication and averaging), the fast fluctuation involving time t is eliminated, but the signal is left with a residual phase:		As shown in Figure 1, a plane wave arrives later at the antenna on the left, so its fluctuating waveform is shifted by a phase proportional to the RF angular frequency times the geometric delay. Here the geometric delay <math>\tau_g</math> is the continuously varying delay required to track the source. The first downconversion by the variable LO inverts the frequencies on both antennas and shifts them by the variable-LO frequency. The second downconversion by the fixed LO inverts and shifts the frequencies by the fixed-LO frequency in a similar manner. This produces an IF frequency in the range 600-1200 MHz. The final downconversion is done by the ADC, which does a final inversion of the frequencies and shifts them by the ADC clock frequency. After digitization, an integer "coarse delay" <math>\tau_1</math>, rounded to the nearest digitizer clock step, is inserted into the right-hand antenna. After correlation (multiplication and averaging), the fast fluctuation involving time t is eliminated, but the signal is left with a residual phase:

	<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \~~tau~~ + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>		<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \tau_1 + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>

	Here <math>\tau_1</math> is the non-integer "fine delay," which must be applied on a channel-by-channel basis across the 600 MHz IF band, while the first term is constant over the band for a given tuning frequency. The phase in equation (2) is to be subtracted from the measured baseline phase.		Here <math>\tau_1</math> is the non-integer "fine delay," which must be applied on a channel-by-channel basis across the 600 MHz IF band, while the first term is constant over the band for a given tuning frequency. The phase in equation (2) is to be subtracted from the measured baseline phase.

Sjyu: /* 1. Background */

2026-03-20T06:34:40Z

1. Background

← Older revision		Revision as of 06:34, 20 March 2026
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	[[File:5.png\|right\|thumb\|600px\|Figure 1: Diagram showing the three downconversions.]]		[[File:5.png\|right\|thumb\|600px\|Figure 1: Diagram showing the three downconversions.]]

	As shown in Figure 1, a plane wave arrives later at the antenna on the left, so its fluctuating waveform is shifted by a phase proportional to the RF angular frequency times the geometric delay. Here the geometric delay <math>tau_g</math> is the continuously varying delay required to track the source. The first downconversion by the variable LO inverts the frequencies on both antennas and shifts them by the variable-LO frequency. The second downconversion by the fixed LO inverts and shifts the frequencies by the fixed-LO frequency in a similar manner. This produces an IF frequency in the range 600-1200 MHz. The final downconversion is done by the ADC, which does a final inversion of the frequencies and shifts them by the ADC clock frequency. After digitization, an integer "coarse delay" <math>tau</math>, rounded to the nearest digitizer clock step, is inserted into the right-hand antenna. After correlation (multiplication and averaging), the fast fluctuation involving time t is eliminated, but the signal is left with a residual phase:		As shown in Figure 1, a plane wave arrives later at the antenna on the left, so its fluctuating waveform is shifted by a phase proportional to the RF angular frequency times the geometric delay. Here the geometric delay <math>\tau_g</math> is the continuously varying delay required to track the source. The first downconversion by the variable LO inverts the frequencies on both antennas and shifts them by the variable-LO frequency. The second downconversion by the fixed LO inverts and shifts the frequencies by the fixed-LO frequency in a similar manner. This produces an IF frequency in the range 600-1200 MHz. The final downconversion is done by the ADC, which does a final inversion of the frequencies and shifts them by the ADC clock frequency. After digitization, an integer "coarse delay" <math>\tau</math>, rounded to the nearest digitizer clock step, is inserted into the right-hand antenna. After correlation (multiplication and averaging), the fast fluctuation involving time t is eliminated, but the signal is left with a residual phase:

	<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \tau + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>		<center><math>\phi = (\omega_{ADC} + \omega_{VLO} - \omega_{FLO}) \tau + \omega_{RF}(\tau_g -\tau_1) \qquad (2)</math></center>