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# Spur Search

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This work was contributed by the Microwaves101 Professor, a true microwave genius who contributes to Microwaves101 from time to time, when he isn't busy fixing his good-old truck or playing in his rock band! There are three parts to this page, and then there is a download that contains the spur-search calculator. Almost all of the text on this page is also included in the spur-search calculator.

### Spur-search background

A spurious response in a mixer or frequency converter is any frequency or range of frequencies that may come into the system, and be converted into the IF band, that is not at the desired or tuned frequency. Spurious responses may be either in or out of the RF input frequency band, although spurious responses that are in the RF input band are much more serious as they cannot be rejected by filtering.

An image signal is an example of a spurious response. All mixers, except for the image-reject type, respond about equally to an "image" signal (RF on the "wrong" side of the LO) and to the "real" signal (RF on the "right" side of the LO). In most systems, frequencies in the image range are filtered out before the first downconversion, or image-reject mixers are used. Another example of a spurious response would be the "2x2" frequency range (in a fundamental, or "1x1" type mixer), where the 2nd harmonic of the RF mixes with the 2nd harmonic of the LO to produce an in-band IF response. Unlike the image response, the "2x2" response and other higher-order responses are typically much lower than the fundamental or desired response in a well-designed mixer. Some mixer manufacturers supply data on the relative responses of "2x2" and other undesired "MxN" frequencies for specific input levels.

Any nonlinear device, when presented with two or more input frequencies, will output not only the input frequencies but harmonics and intermodulation products of the input frequencies as well. The expression normally used for the mixing products is:

F

_{IF}= M * F_{RF}+ N * F_{LO }(1)

Where M and N are equal to 0, ±1, ±2, etc. In mixers, typical values of M and N for the "desired" response are M = 1 and N = -1 (for fundamental mixing with low-side LO injection), M = -1 and N = 1 (for fundamental mixing with high-side LO injection), and M = 1 and N = -2 (for subharmonic mixing with low-side LO injection). Although there is usually only one value each for M and N for the desired response, there are many more values of M and N that can produce undesired, or spurious, responses. Some examples:

M (RF harmonic) Value | N (LO Harmonic) Value | Signal |
---|---|---|

1 | -1 | Desired RF Input Signal |

-1 | 1 | Image Frequency |

1 | 0 | RF to IF Leakage |

0 | 1 | LO to IF Leakage |

2 | -2 | Undesired (2x2) Response |

-2 | 2 | Another 2x2 Response |

RF to IF leakage and LO to IF leakage are not technically spurious responses, since they are not in the IF band. Note that the definitions of the desired input signal versus the image or other spurious responses depends on the receiver design. Using this terminology, spurious responses can be defined as all responses (that produce outputs that fall in the IF band) for which M (the RF harmonic) and N (the LO harmonic) are not equal to the desired values.

Some rules of thumb for spurious responses can be derived from expression (1) above, when the desired values of M and N have a magnitude of one (fundamental mixing) and the undesired M and N are equal in magnitude (e.g., M = 2 and N = -2). The first is that the input frequency or range of frequencies that will produce an MxM response is any frequency for which the fundamental IF response would be equal to the desired IF frequency divided by M. Stated mathematically,

F

_{RF}(Mxm) = F_{RF}F_{IF}(Fund.) = F_{IF}(Desired)/M (2)

For example, any RF input frequency whose fundamental IF response would be half the desired IF frequency would constitute a 2x2 response. An RF input frequency whose fundamental IF response would be one-third the desired IF frequency would constitute a 3x3 response.

From this rule, a second rule can be derived. This rule specifies the maximum IF center frequency that will lead to in-band Mxm spurs. This depends on both the LO bandwidth (tuning range) and the IF bandwidth. The rule, stated mathematically, reads:

F

_{C|IF,MAX.}(Mxm) = M/(M-1) * BW_{LO}+ (M+1)/(2M - 2) * BW_{IF }(3)

You can also look at this as a lower bound on a usable IF center frequency that will NOT support in-band Mxm spurs. This would read:

F

_{IF,CEN.}(Mxm) > M/(M-1) * BW_{LO}+ (M+1)/(2M - 2) * BW_{IF}(4)

For example, to move all 3x3 spurious responses out of the input RF band, the IF center frequency would have to be greater than 3/2 (=1.5) times the LO bandwidth, plus 4/4 (=1.0) times the IF bandwidth. To move all 2x2 spurious responses out of the input RF band, the IF center frequency would have to be greater than twice the LO bandwidth, plus 3/2 times the IF bandwidth.

Since the term that multiplies the IF bandwidth is always smaller than the term that multiplies the LO bandwidth for any M value, an approximation to this rule of thumb that's always safe to use is:

F

_{IF,CEN. }(Mxm) > M/(M-1) * BW_{RF}(5)

Since the RF bandwidth is always equal to the LO bandwidth plus the IF bandwidth for fundamental mixers. For example, to suppress all 3x3 spurious responses (move them out of the RF band), the IF center frequency should be greater than 3/2 the RF bandwidth. To suppress all 2x2 spurious responses, the IF center frequency needs to be greater than twice the RF bandwidth. Since this rule always guarantees that Mxm spurious responses are outside the RF band, and it's simpler to remember than (4), use this one unless constraints require the use of (4). This would probably only happen for fixed-LO, wideband IF receivers, where BW_{LO} is zero or small compared to BW_{IF}.

Input frequencies that generate MxN spurs, where |M| |N|, depend on the ratios of the LO and IF frequencies as well as the bandwidths. Thus, there are no simple rules of thumb such as (5) above to prevent these spurious responses. To identify these types of responses, a spur finder program is needed. That is the purpose of this workbook.

### Continuos LO range

Enter the range of LO frequencies that the system will use in the "LO Frequency Range" cells, K6 and M6. Put the lowest LO frequency in cell K6 and the highest LO frequency in cell M6. Enter the IF frequency range in cells K7 and M7, with the lowest IF frequency in cell K7 and the highest in cell M7. Then, enter the M (RF harmonic) and N (LO harmonic) values that the system (or the conversion stage with which you are concerned) uses into cells K8 (for M) and K9 (for N). For example, for fundamental mixing with low-side LO injection, use M = 1 and N = -1. For fundamental mixing with high-side LO injection, use M = -1 and N = 1. For subharmonic mixing with low-side LO injection, use M = 1 and N = -2. The worksheet will then automatically compute the RF frequency range and display it in the area labeled "Full RF Frequency Range". Use these numbers as a sanity check to make sure you've entered the correct LO frequency range and IF bandwidth. Remember that, for a system using fundamental mixing, the RF bandwidth is the sum of the LO and IF bandwidths. Enter the range over which you want the worksheet to search for spurs in the "Spurious Response Search Range", cells K11 and M11. This range should normally be somewhat wider than the "Full RF Frequency Range". For example, if the system is exposed to interfering frequencies just outside of the "Full RF Frequency Range" and uses an RF bandpass filter ahead of the downconverter, you may want to enter the bandwidth of this filter as the "Spurious Response Search Range".

Now, check the two frequency tables for cells with bold type. You should see your RF frequency range in a pair of cells with bold type and a green background. The frequencies in these cells will match those in the "Full RF Frequency Range" area. Next, look for cells with bold type and a yellow or red background. Cells with a yellow background will display spurious responses in the "Spurious Response Search Range" of frequencies. The minimum and maximum frequencies shown in these cells represent the range of input frequencies that will downconvert to frequencies in the specified IF frequency range, somewhere in the tuning range of LO frequencies. Cells with a red background display spurious responses in the "Full RF Frequency Range". These cells are highlighted with a red background because they display frequencies that are inside the computed RF band, which means they can't be filtered out before the downconversion. Any cells that contain bold type and yellow or red backgrounds represent spurious responses in the downconverter. For these cells, check the M (RF harmonic) and N (LO harmonic) values. The M values correspond to the row number in the table, and the N values correspond to the column number. You can use the values of M and N to get an idea of the severity of the spurious response. For example, a high-order spurious response (e.g., M = 8 and N = -9, or "8 x 9") would have a much lower output from the converter than a low-order spur such as M = 2 and N = -2, or "2 x 2". Also, the RF order, or M value, of a spurious response dictates how the spurious output level will vary as a function of the input level. For example, a spur with an M value of 2 (or -2) will cause an IF output that increases by 2 dB for each 1 dB increase in RF input level. A spur with an M value of 3 (or -3) will have an output level increase of 3 dB for each 1 dB increase in RF input level. Some mixer manufacturers provide tables of spurious response levels for specific levels of LO and RF input power. If you want to have the worksheet search for spurs with M and/or N values above 9, you can change the "M" values (in rows 18 and 32) and the "N" values (in column D). The only thing you have to watch out for is this: you need to keep the "M" values positive and the "N" values negative in Table I, and keep the "M" values negative and the "N" values positive in Table II. Otherwise, the frequency ranges in the tables won't be right.

Why are there two frequency tables? Because, just like the desired response, the spurious responses have images. In fact, the image frequency range itself is technically a range of spurious responses. For example, for the case of fundamental mixing with low-side LO injection (M = 1 and N = -1), the first row and column of Table I represent the desired range of input frequencies, and will be highlighted with a green background. The first row and column of Table II will then represent the image frequency range. If this range of frequencies is inside your "Spurious Response Search Range", or somehow, in your "Full RF Frequency Range", then one or both of the frequencies will be highlighted with a yellow or red background.

The main purpose of the "Continuous LO Range" worksheet is to show if you've got any spurious responses at all, and what their frequency range and order (M and N values) are. To find out where in your system's frequency band you would tune to in order to be susceptible to the spurious responses, use the "Discrete LO Channels" worksheet.

### Discrete LO range

Enter the range of LO frequencies that the system will use in the "LO Frequency Range" cells, H4 and J4. Put the lowest LO frequency in cell H4 and the highest LO frequency in cell J4. Enter the LO frequency step size (channel spacing) in cell M4. Enter the IF frequency range in cells H7 and J7, with the lowest IF frequency in cell H7 and the highest in cell J7. Then, enter the M (RF harmonic) and N (LO harmonic) values that the system (or the conversion stage with which you are concerned) uses into cells J8 (for M) and J9 (for N). For example, for fundamental mixing with low-side LO injection, use M = 1 and N = -1. For fundamental mixing with high-side LO injection, use M = -1 and N = 1. For subharmonic mixing with low-side LO injection, use M = 1 and N = -2. The worksheet will then automatically compute the RF frequency range and display it in the area labeled "Full RF Frequency Range". Use these numbers as a sanity check to make sure you've entered the correct LO frequency range and IF bandwidth. Remember that, for a system using fundamental mixing, the RF bandwidth is the sum of the LO and IF bandwidths. The worksheet will also display the tuned RF frequency range (a function of the slider position and the channel number), and the tuned RF center frequency. Enter the range over which you want the worksheet to search for spurs in the "Spurious Response Search Range", cells K11 and M11. This range should normally be somewhat wider than the "Full RF Frequency Range". For example, if the system is exposed to interfering frequencies just outside of the "Full RF Frequency Range" and uses an RF bandpass filter ahead of the downconverter, you may want to enter the bandwidth of this filter as the "Spurious Response Search Range".

Now, check the two frequency tables for cells with bold type. You should see your RF frequency range in a pair of cells with bold type and a green background. The frequencies in these cells will match those in the "Tuned RF Frequency Range" area. Next, look for cells with bold type and a yellow or red background. Cells with a yellow background will display spurious responses in the "Spurious Response Search Range" of frequencies. The minimum and maximum frequencies shown in these cells represent the range of input frequencies that will downconvert to frequencies in the specified IF frequency range, with the current LO frequency. Cells with a red background display spurious responses in the "Full RF Frequency Range". These cells are highlighted with a red background because they display frequencies that are inside the computed RF band, which means they can't be filtered out before the downconversion. Any cells that contain bold type and yellow or red backgrounds represent spurious responses in the downconverter. For these cells, check the M (RF harmonic) and N (LO harmonic) values. The M values correspond to the row number in the table, and the N values correspond to the column number. You can use the values of M and N to get an idea of the severity of the spurious response. For example, a high-order spurious response (e.g., M = 8 and N = -9, or "8 x 9") would have a much lower output from the converter than a low-order spur such as M = 2 and N = -2, or "2 x 2". Also, the RF order, or M value, of a spurious response dictates how the spurious output level will vary as a function of the input level. For example, a spur with an M value of 2 (or -2) will cause an IF output that increases by 2 dB for each 1 dB increase in RF input level. A spur with an M value of 3 (or -3) will have an output level increase of 3 dB for each 1 dB increase in RF input level. Some mixer manufacturers provide tables of spurious response levels for specific levels of LO and RF input power. If you want to have the worksheet search for spurs with M and/or N values above 9, you can change the "M" values (in rows 18 and 32) and the "N" values (in column D). The only thing you have to watch out for is this: you need to keep the "M" values positive and the "N" values negative in Table I, and keep the "M" values negative and the "N" values positive in Table II. Otherwise, the frequency ranges in the tables won't be right.

Why are there two frequency tables? Because, just like the desired response, the spurious responses have images. In fact, the image frequency range itself is technically a range of spurious responses. For example, for the case of fundamental mixing with low-side LO injection (M = 1 and N = -1), the first row and column of Table I represent the desired range of input frequencies, and will be highlighted with a green background. The first row and column of Table II will then represent the image frequency range. If this range of frequencies is inside your "Spurious Response Search Range", or somehow, in your "Full RF Frequency Range", then one or both of the frequencies will be highlighted with a yellow or red background.

The main purpose of the "Discrete LO Channels" worksheet is to identify which channels in your system are susceptible to spurious responses. You can move the slider across to adjust the LO frequency (and thus the frequency the system is tuned to and the channel number) and watch the changes to the spurious response tables. To find all the spurious responses of the system in a single view, use the "Continuous LO Range" worksheet.

Good luck!

**Author : **Unknown Editor

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