The Distances Chart: A New Approach to Spurs Calculation

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When dealing with frequency conversion processes, it is necessary to know the type and level of the non-desired spurs appearing at the output of the frequency conversion device. Power level calculations rely on mixer behavior (that is the nonlinear transfer function from V_in to V_out) that is dependant on specific mixer technology and design.

These levels are commonly referred to the level of the wanted output signal and are expressed as rejections in decibels (positive values) or gains (negative values). Analytical expressions for intermodulation product (IMP) suppressions can be found1 for the case of doubly balanced diode ring mixers; they take into account the balun imbalances, diode mismatches and diode turn-on voltages. The proper setting of these parameters can require a deep knowledge of the internal mixer circuitry and technology, not often available.

The rejections to IMPs are also dependant on the relative levels of oscillator and input signals and they vary with frequency. Many manufacturers of microwave mixers provide spur rejection tables in their datasheets, corresponding to conversions between a mid-to-high RF and a low IF, usually of 100 MHz. In conversions to/from much higher intermediate frequencies, as those considered in this study, it may be necessary to obtain new spur tables from measurement, prior to their use in a microwave simulator. Also in a first approach, the analytical results from Henderson¹ could be used.

Regardless of the output spectrum component levels in a particular frequency conversion, their frequency values can be obtained in a much more generic way: All the convolutions between harmonics of the oscillator and input signals will appear at the mixer output. Thus, for single tone RF and LO inputs to a mixer, the IF output will consist of a set of spectral lines with positive frequencies expressed as |mRF ± nLO|, that will be named |IF_mn| spurs. For the three common frequency conversion types—RF+LO, LO-RF and RF-LO—the desired output signals correspond to IF₁₁, IF_-11 and IF_1-1.

The Conversion Frequency Plan

The proper choice of IF and LO values can greatly simplify the filtering requirements in a converter design, produce high robustness to interferences in receiver applications, or high output spectral purity in transmitters, and allow for a reduced number of filters and minimum values of local oscillator frequencies, while covering a broad RF band with a wide IF bandwidth. These aspects become particularly important in converter applications requiring wide IF bandwidths and broad RF bands, like in multiband and multistandard transceivers and SDRs.²

Figure 1 shows the datasheet spur table for the mixer HMC141 from Hittite Microwave Corp. This table represents the mixer rejection to the feedthrough harmonics (the nLO (m = 0) and mRF (n = 0) signals) and the IMPs of the form |mRF + nLO|. The suppressions are given in dBc below the desired output signal (equal to 0 dBc). The test conditions are single tone RF and LO and conversion to low IF (100 MHz in this case). The spur tables are always referred to specific input and LO signal power levels.

An optimum frequency plan for the conversion represented by the above table will keep the low order IMPs (those with m ≤ 2 and n ≤ 4) away from the IF band to be rejected by filtering, and will leave only high order spurs (m ≥ 3) appearing at the desired output band as they are well rejected by the mixer. This is a necessary limitation if broadband RF filtering is to be implemented in order to reduce the filter count, and its direct consequence is that the mixer rejections to those spurs must be equal to or higher than the dynamic range specified for the converter. This imposes an important requirement for mixer selection.

The Classical Spurs Chart Method

The Spurs Chart is a plot, on the IF/RF plane, of the mixer output frequencies as a function of the single input RF tone frequency for a fixed LO value. The |mRF+nLO| products appear as straight lines, crossing the y-axis (IF) at ±nLO with a slope ±m in the x-axis (RF); (m, n) being integer numbers m1 and n0.

In order to study the spurs appearing at the mixer output when the LO is swept to translate into IF a portion of the input band in a receiver with broad RF filtering (or to translate the IF into the broad RF band in a transmitter), several steps must be performed trying different values for LO, IF and the RF filter pass band, until the most harmful IMPs are kept outside the mixer output and at a safe distance that will make their rejection effective by practical filtering.

The former process can be tedious and inefficient as the represented spur lines change for every LO setting and many different frequency plans are possible, making it very difficult to find the optimum values for the RF bandwidth, LO band and the IF. An example for an LO sweep is given in Figure 2, where the desired output is IF = LO – RF (in blue). The higher order IMPS are plotted in grey.

The Spurs Chart results are interpreted as follows: The range of RF frequencies defined by the LO-RF line crossing the IF band are the selected RF sub-band (encircled box in Figure 2) and is dependant of the IF and LO values chosen. There is no single solution for the IF and LO bands unless the additional constraint of having the maximum possible RF filter band is imposed.

The chart also shows that many other lines cross the IF band; these IMPs are spurs originated from the LO mixing with non-wanted input RF frequencies outside the RF filter band. It is the task of the RF input filter to eliminate these frequencies, or reduce them enough to make them appear, at the mixer output, at a much lower level than the wanted IF.

Other lines cross IF at the selected RF sub-band; they are IMPs of the wanted input and the LO that, through a different conversion other than LO-RF, produce the same IF output. In order to prevent their signal distortion effect they must fall in the rejection band of the IF filter or be properly rejected by the mixer. A practical IF filter will have its rejection and pass bands separated by a “guard” distance that, together with the rejection requirements, will impose the type and order of the filter.

The maximum pass band for the RF filter will be determined by its guard value selected (not shown in the chart) and the IMPs allowed in the IF band. In this example, the low order IMPs having m ≤ 2 and n ≤ 4 will be rejected before entering the mixer, in order not to appear at the IF band.

Each time the LO is modified, the plotted lines shift from their position and new RF frequencies and spurs may appear at the IF band. It is then difficult to work with this type of chart when the initial IF and LO frequencies are unknown and the LO is going to be swept. One can get lost easily, dealing with all the lines moving as IF and LO change. It is also difficult to follow the evolution of a particular spur and see its dependency with the values of the frequency plan during a trial-and-error optimization.

An extension of the traditional Spurs Chart has been proposed3 to calculate the band segments of incidence of the spurs within a range of LO frequencies, thus solving the problem of LO static. The spur lines are observed within a 3D-like figure bounded by two apertures or boxes (each one associated with LO_min and LO_max, similar to the encircled boxes in Figure 2) and their connecting lines.

This article presents a new approach called the Distances Chart, which also solves the drawbacks associated with the LO static and makes more comfortable the study of the conversion spurs in applications where a broad RF band is to be swept with a wide IF.

The New Method Description

The distance from the RF input frequency to the RFmn spur is defined as

The intermodulation products at the mixer output are calculated as |mRF + nLO| or just mRF + nLO if (m, n) are allowed to take positive and negative values; the IMPs falling at negative frequencies are also considered. By definition, the RF_mn interferences verify the following condition:

where IF is the center of the output band. Extracting d_mn from the above expressions gives

The distance from the n^thLO harmonic to the limit of the IF band is

Finally, if IF = LO - RF is selected, one obtains

Thus, d_mn (y-axis) can be plotted against RF (x-axis), having IF, Guard and IF_BW as the design parameters, and no dependency with the LO sweep (it is already implicit in the result).

The design approach for the frequency plan is to find the widest RF filters—in order to minimize their number—having good rejections to the RF_mn frequencies. The minimum d-_mn and d+_nm distances from every in-band frequency to the closest RF_mn interferers must be determined. The constraints for d_nm are shown in Figure 3:

with rf being the center frequency of a selected sub-band, IF_BW wide, passing through the RF filter pass band.

At the lower end of the RF filter band is rf1 = RF_min and the constraints are

At the top end of the RF filter band is rf2 = RF_max and the constraints become

“Guard” is set by trial and error and its value affects directly the design of the RF and IF filters; the higher the Guard value, the softer the RF/IF filter specifications, but the RF sweep band is reduced. The spur-free area at the Distances Chart is shown in Figure 4.

The IF values are optimized for spur rejection by combining the rejection capabilities of the mixer to high order intermodulation products or spurs, and the IF filter rejection to the low order intermodulation products (poorly rejected by the mixer), provided they appear at a safe distance away from the IF band. IF = LO-RF is the preferred conversion mode because it allows for lower IF values (simplifying the next conversion stages) and better rejection of the LO harmonics in transmitters, as none of them can fall inside the broad RF output band.

Depending on the LO sweep selected, 1 to 3 different IF values are required in the first conversion stage of a receiver (or the last stage in the Transmitter case) to make a detour of the otherwise unavoidable spurs. If IF is high enough, only one single value can satisfy the condition of rejecting the lower order IMPs from the mixer, but this also implies high values for the LO, which will be in detriment to its price and phase noise performance.

Case Study: Development of an optimum frequency plan for a 0.22 to 6 GHz RF band to a 100 MHz wide IF with 70 dB SFDR

For the purpose of this example, a mixer with the spur table shown in Figure 1 will be considered. It will be assumed, without losing generality, that the rejection values are valid for all the RF, LO and IF frequencies involved. In a real case, an accurate nonlinear model for the mixer would be needed or different spur tables shall be obtained through a device characterization process, but such a study is beyond the scope of this article. The Distances Chart is intended for qualitative results only, and can be of great help in the very first phases of a converter design, during the definition of its topology and the specification of requirements for its main components.

The frequency plan, implemented with the lowest IF value and LO sweep, uses three different IFs for the same conversion stage and is presented in Figure 5. The whole 1.33 to 6.07 GHz RF band can be covered with only three wide pass band RF filters while guaranteeing the rejection to harmful interferers (those falling at or close to IF). The problem of filtering out the lower RF interferers will be discussed at the end.

Figures 6 to 8 show the Distances Charts results for the 1.33 to 6.07 GHz RF portion of the frequency plan that keeps the low order IMPs (those with m≤2 and n≤4) at a minimum “Guard” distance away from the 100 MHz wide IF band.

Figure 6 is the conversion of the 4.30 to 6.07 GHz RF band to a 100 MHz wide IF centered at 4.03 GHz using a single RF filter and keeping the most harmful IMPs at a Δf ≥ 215 MHz from the IF band; Figure 7 is the conversion of the 2.49 to 4.40 GHz RF band to a 100 MHz wide IF centered at 4.68 GHz using a single RF filter and keeping the most harmful IMPs at a Δf ≥ 215 MHz from the IF band; Figure 8 is the conversion of the 1.33 to 2.59 GHz RF band to a 100 MHz wide IF centered at 5.50 GHz using a single RF filter and keeping the most harmful IMPs at a Δf ≥ 60 MHz from the IF band.

In the lower RF frequency band (below 1.3 GHz) the LO-2RF product (identified in Figure 6) can be very harmful if an interferer appears at half the wanted signal frequency (LO-2RF_-21 = LO-RF_-11 = IF). This is not dependant on the frequency plan selected.

The interferer proximity to the wanted signal (only ½ RF_-11 separation) will prevent the use of a broadband RF filter if the mixer rejection to the 2 X 1 product is lower than the spur-free dynamic range required in the application (see the grey colored box in the table of Figure 1). Four mid-to-narrow pass band RF filters would be required here to cover the 0.22 to 1.44 GHz range (see Figure 9). The use of a mixer with higher rejection to the 2 X 1 product will reduce the RF filter constraints.

Even when no interference is present at the RF port, there is always the LO-2RF product falling at an RF distance from the IF in Rx operation. This limits the rejection capabilities of the IF filter, or puts a hard constraint on its design; both aspects degrade the overall performance and increase costs. Narrow band RF filters would also be required for Tx operation in order to guarantee a high spur-free dynamic range at the RF side, also increasing complexity and cost.

The spurs table of a given mixer can be modified if input and LO levels different than the ones used during table extraction are allowed. The dependency of the spur table rejections with the input and LO signals power levels can be approximated in double-balanced mixers¹ by

where m is the order of the input signal and dBIN, dBLO represent the variation of input and LO levels from the reference values used during Spur Table generation. This can be used to increase the mixer rejection to problematic spurs and reach the required SFDR, but the cost is a reduction in the output dynamic range. Example: A mixer presenting a 64 dB rejection to the 2 1 spur measured at –10 dBm RF will benefit from a 74 dB rejection, if fed with a –20 dBm input (for the same LO power).

Mixing products with m ≥ 3rd order for the RF and n ≥ 5th order for the LO have not been considered in the Distances Chart implemented for this study. This is a necessary limitation if broadband RF filtering is wanted, and its direct consequence is that the mixer rejections to these spurs must be equal or higher than the dynamic range specified for the converter, in case they could fall too close to or even inside the IF band.

Commercially available MMIC mixers show rejections higher than 70 to 80 dBc to these higher order IMPs, but many of those mixers do not show high enough rejections to the 3LO-3RF₃₃ (= IF) product. If one of those mixers was used, the frequency plan shall be set to avoid RF₃₃ = LO - 1/3IF. Then RF₃₃ > RFmax would become a new constraint, and an additional requirement for the frequency plan shall be

Conclusion

The Distances Chart is a new method to find mixer spurs, allowing for an easy optimization of the IF and RF filter bands as well as the LO sweep band in applications where a broad frequency band with wide IF is to be covered, as in the case of broadband double conversion transceivers for multiband, multistandard and SDR applications. This method is easier and more straightforward to use than the traditional spurs chart approach during the study phase of such converters.

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Acknowledgment

This work has been developed within the scope of the TelMAX Project and is partially funded by Centro para el Desarrollo Tecnológico e Industrial (CDTI) of the Spanish Ministry of Science and Innovation, under the INGENIO 2010 Program/CENIT call.

References
1. B. Henderson, “Predicting Intermodulation Suppression in Double-balanced Mixers,” Watkins-Johnson Co. Tech-notes, Vol. 10, No. 4, July/August 1983.
2. TelMAX Project (ref. CEN20071036), http://www.proyectotelmax.com/.
3. D. Gandhi and C. Lyons, “Mixer Spur Analysis with Concurrently Swept LO, RF and IF: Tools and Techniques,” Microwave Journal, Vol. 46, No. 5, May 2003, pp. 212-220.

José Luis Flores graduated as a Tele-communications Engineer in 1995 from the UPC-Universitat Politècnica de Catalunya-in Barcelona and was granted with a traineeship at ESA-ESTEC in The Netherlands, where he started his career as a MMIC designer. He later worked in France for Alcatel-LEMMIC, OMMIC and Alcatel-Opto+, mainly focusing on the application of GaAs-pHEMT and InP-HBT technologies for satellite onboard communication systems and ultra-wideband fiber-optic links. In 2000 he followed the Summer Session Program of the International Space University. He has also participated in two scientific space missions: TEAMSAT and NANOSAT, both launched by Ariane-V in 1997 and 2004, respectively. Since 2005, he has been with AT4 Wireless in Málaga, where he is working on the design of broadband reconfigurable transceivers and high accuracy microwave power detectors.