Patent Publication Number: US-9835677-B2

Title: Method and system for producing a signal with a power change determined by a phase or frequency difference between two signal sources

Description:
BACKGROUND 
     Linearity is an important performance characteristic of many electronic devices. The linearity of a device may be defined as the degree to which the amplitude (or power) level of an output signal of the device is directly proportional to the amplitude (or power) level of an input signal provided to the device. 
     The slope of a plot of a device&#39;s output signal level versus the device&#39;s input signal level may be referred to as the device&#39;s gain. For many types of devices, ideally the device should have a perfectly linear response wherein the output signal level changes perfectly linearly in response to a change in the input signal level, yielding a constant gain. 
     However actual devices do not have perfectly linear responses, and thus there is some variation in the gain of the device as a function of one or more parameters. In many cases, it is desired to be able to determine and specify the linearity of a device as a function of some parameter (e.g., input signal level, frequency, input power level, temperature, input voltage level, etc.). Linearity may be specified in a number of different manners, but one common way to specify linearity is to identify the deviation of the device&#39;s actual output-signal-level-versus-input-signal-level response from a straight line over a given range of a particular parameter. Where a device&#39;s gain, G, is specified in decibels (dB), the device&#39;s linearity may be specified as the amount in dB that the actual output-signal-level-versus-input-signal-level varies from G. To illustrate, when an example amplifier has a nominal gain of 20 dB, the amplifier&#39;s linearity may be specified, for example, as “±0.50 db over an input signal level range from −80 dbm to −20 dbm.” This is just one example manner of specifying a device&#39;s linearity which is provided for illustration purposes, and many other ways of specifying linearity are known to those of skill in the art. 
     One category of devices for which linearity may be an important performance characteristic includes detectors. As broadly defined here, a detector is a device which receives a receive signal and which, in response to the receive signal, outputs an output signal whose signal level may linearly track the signal level of the input signal. Examples of detectors include, without limitation, crystal detectors, diode detectors, amplifiers, mixers, down-converters, analog-to-digital converters (ADCs), and power meters. 
       FIG. 1A  illustrates an arrangement  10  for measuring the linearity of a device-under-test (DUT)  50 . The arrangement  10  includes a signal generator  15 , a reference attenuator  20  and a processor  40 . Of significance, in arrangement  10  reference attenuator  20  has a calibrated or otherwise predetermined attenuation characteristic, and attenuation characteristic data for reference attenuator  20  is available to processor  40  (e.g., stored in a memory accessible by processor  40 ). In some alternative arrangements, reference attenuator  20  could be replaced by a reference amplifier or other variable gain device with a calibrated or otherwise predetermined gain characteristic. 
     Arrangement  10  may be employed to determine the linearity of DUT  50  by changing an attenuation value of reference attenuator  20  to thereby change the power level supplied to DUT  50 . Processor  40  may then compare the corresponding change in the output signal level produced by DUT  50  to the change in attenuation provided by reference attenuator  20  based on the attenuation characteristic data for reference attenuator  20 . This measurement may be repeated for a number of different attenuation values in order to determine the linearity of DUT  50 . 
       FIG. 1B  illustrates another arrangement  11  for measuring the linearity of a DUT  50  that does not require a calibrated reference attenuator. The arrangement  11  includes a signal generator  15 , an attenuator  25 , a power splitter  30 , a linear reference detector  35 , and a processor  40 . Of significance, in arrangement  11  linear reference detector  35  has a calibrated or otherwise predetermined linearity characteristic, and the linearity characteristic data for linear reference detector  35  is available to processor  40  (e.g., stored in a memory accessible by processor  40 ). 
     Arrangement  11  may be employed to determine the linearity of DUT  50  by changing an attenuation value of attenuator  25  to thereby change the power level supplied to both DUT  50  and linear reference detector  35  by the same amount. Processor  40  may then compare the corresponding changes in the output signal levels produced by DUT  50  and linear reference detector  35  in response to the change in their input signal levels, together with the linearity characteristic data for linear reference detector  35 , in order to determine the linearity of DUT  50 . In arrangement  11 , attenuator  25  does not need to be calibrated or have a known attenuation response because whatever the actual attenuation change it provides between two different attenuator settings, the same attenuation in signal level is provided to both DUT  50  and linear reference detector  35 . 
     The arrangements and techniques described above have some drawbacks, notably with respect to speed and accuracy. For example, in some cases, linearity tests must be able to determine the linearity of devices which employ high speed (e.g., 10 megasamples/second) analog-to-digital converters with more than 14 bits of resolution with quantization error reduction. In these cases, a very high degree of speed and accuracy is required. However, linear reference detectors such as that employed in arrangement  11  commonly require long measurement settling times. Also, arrangements  10  and  11  both depend upon the use of characteristic data for a calibrated or reference device (e.g., reference attenuator  20  or linear reference detector  35 ). So any change in this behavior after the reference device has been calibrated or characterized, or any error in characterizing the reference device, results in an inaccuracy or uncertainty in the linearity measurement of DUT  50 . 
     Accordingly, it would be desirable to provide a method and system for determining the linearity of a device under test which does not depend upon having a calibrated reference attenuator or detector. It would also be desirable to provide a method and system for determining the linearity of a device under test which can avoid the long settling times associated with the use of linear reference detectors. 
     SUMMARY 
     In an example embodiment, a method comprises: combining a first periodic signal and a second periodic signal to produce a combined signal, wherein the second periodic signal has at least one of a phase difference and a frequency difference with respect to the first periodic signal; applying the combined signal to the input of a device; and determining a linearity of the device from an output signal of the device based on the at least one of the phase difference and frequency difference between the first periodic signal and the second periodic signal. 
     In another example embodiment, a system comprises: a first signal generator configured to output a first periodic signal; a second signal generator configured to output a second periodic signal, wherein the second periodic signal has at least one of a phase difference and a frequency difference with respect to the first periodic signal; a signal combiner configured to combine the first periodic signal and a second periodic signal to produce a combined signal and to supply the combined signal to a device; and a processor configured to determine a linearity of the device from an output signal of the device based on the at least one of the phase difference and frequency difference between the first periodic signal and the second periodic signal. 
     In yet another example embodiment, a system is configured to produce a signal with power change. The system comprises a first source and a second source, and the power change is determined by the one of a phase difference or a frequency difference between the first and second sources. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The example embodiments are best understood from the following detailed description when read with the accompanying drawing figures. In fact, the dimensions may be arbitrarily increased or decreased for clarity of discussion. Wherever applicable and practical, like reference numerals refer to like elements. 
         FIGS. 1A-B  illustrate two different arrangements for measuring the linearity of a device. 
         FIG. 2  illustrates one example embodiment of an arrangement for measuring the linearity of a device-under-test (DUT) using two periodic signals. 
         FIG. 3  is a flowchart of one embodiment of a method of determining the linearity of a DUT using two periodic signals. 
         FIG. 4  illustrates one example embodiment of an ideal output response of a DUT in the arrangement of  FIG. 2 . 
         FIG. 5  illustrates another example embodiment of an arrangement for measuring the linearity of a DUT using two periodic signals. 
         FIG. 6  illustrates another example embodiment of an ideal output response of a DUT in the arrangement of  FIG. 2 . 
         FIG. 7  illustrates yet another example embodiment of an arrangement for measuring the linearity of a DUT using two periodic signals. 
         FIG. 8  illustrates an example embodiment of an ideal output response of a DUT in the arrangement of  FIG. 7  for various attenuation settings. 
         FIG. 9  compares examples of the mean error and repeatability &amp; reproducibility variation of linearity measurements made using an example embodiment of an arrangement as illustrated in  FIG. 7 , and an example of an arrangement as illustrated in  FIG. 1B . 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, for purposes of explanation and not limitation, example embodiments disclosing specific details are set forth in order to provide a thorough understanding of an embodiment according to the present teachings. However, it will be apparent to one having ordinary skill in the art having had the benefit of the present disclosure that other embodiments according to the present teachings that depart from the specific details disclosed herein remain within the scope of the appended claims. Moreover, descriptions of well-known apparati and methods may be omitted so as to not obscure the description of the example embodiments. Such methods and apparati are clearly within the scope of the present teachings. 
     Unless otherwise noted, when a first device is said to be connected to a second device, this encompasses cases where one or more intermediate devices may be employed to connect the two devices to each other. However, when a first device is said to be directly connected to a second device, this encompasses only cases where the two devices are connected to each other without any intermediate or intervening devices. Similarly, when a signal is said to be coupled to a device, this encompasses cases where one or more intermediate devices may be employed to couple the signal to the device. However, when a signal is said to be directly coupled to a device, this encompasses only cases where the signal is directly coupled to the device without any intermediate or intervening devices. 
       FIG. 2  illustrates one example embodiment of an arrangement  200  for measuring the linearity of a device-under-test (DUT)  50  using two periodic signals. The arrangement  200  includes: a first signal generator (or first source)  210 , configured to output a first periodic signal S 1 ; a second signal generator (or second source)  215  configured to output a second periodic signal S 2 ; a signal combiner  230  configured to combine the first periodic signal (e.g., a sinusoidal signal) S 1  and the second periodic signal (e.g., a sinusoidal signal) S 2  to produce a combined signal S 3 , and to supply the combined signal S 3  to DUT  50 ; and a processor  240  configured to receive an output signal S 4  from DUT  50 . 
     For simplicity of explanation, in the description to follow it is assumed that first periodic signal S 1  and second periodic signal S 2  are each sinusoidal signals, and they will be referred to hereafter as first sinusoidal signal S 1  and second sinusoidal signal S 2 . 
     In a beneficial arrangement, as indicated by the bidirectional arrow in  FIG. 2 , first signal generator  210  and second signal generator  215  may be phase coherent with each other, meaning that their respective first and second sinusoidal signals S 1  and S 2  have a controlled relationship in frequency and phase with respect to each other. That is, second sinusoidal signal S 2  may be offset in phase or frequency with respect to first sinusoidal signal S 1 , but the phase offset or frequency offset between first and second sinusoidal signals S 1  and S 2  does not drift to a significant degree over relevant time period for making a linearity measurement of DUT  50 . 
     It is desirable that S 3  be a linear combination of first and second sinusoidal signals S 1  and S 2 , such that the second sinusoidal signal S 2  does not appear on the port of signal combiner  230  connected to first signal generator  210 , and that the first sinusoidal signal S 1  does not appear on the port of signal combiner  230  connected to second signal generator  215 . This prevents an interaction between first and second signal generators  210  and  215  that could change the power level that either signal generator transmits when the other signal generator is turned off. Therefore, in a beneficial feature, signal combiner  230  is an isolating signal combiner that provides signal isolation between its two input ports and their associated first and second signal generators  210  and  215 . The level of isolation required to be provided by signal combiner  230  depends on the desired level of linearity measurement accuracy, and the dynamic range over which the linearity measurement is to be performed. 
     Processor  240  may be any kind of device that is capable of determining the linearity of DUT  50  based on the output signal S 4  from DUT  50  according to any of the various embodiments of methods described below. 
     As described in greater detail below, in various embodiments processor  240  is configured to determine the linearity of DUT  50  from an output signal of DUT  50  based on a phase difference or a frequency difference between a component of the first periodic signal S 1  present in the output signal S 4  and a component of the second periodic signal S 2  present in the output signal S 4 . In various embodiments, processor  240  may comprise: a central processing unit (CPU) or general purpose processor executing a set of instructions in accordance with software code stored in a memory device; a digital signal processor; an application specific integrated circuit (ASIC), a programmable gate array device, for example a field programmable gate array (FPGA); or any appropriate combination of hardware, firmware, and software. 
     The arrangement  200  may be employed by various embodiments of methods for determining the linearity of DUT  50 . As described below, various embodiments eliminate reliance upon the linear behavior of a reference device such as a reference detector. In various embodiments, the power levels of the first and second sinusoidal signals S 1  and S 2  presented to signal combiner  230  are each held constant, and the power level of combined signal S 3  presented to DUT  50  is not changed by a variable attenuator or other variable gain device, but instead by a mathematically predictable pattern based on the summation of the first and second sinusoidal signals S 1  and S 2  having a defined phase or frequency difference between them. 
       FIG. 3  is a flowchart of one embodiment of a method  300  of determining the linearity of a DUT using two periodic signals. In some embodiments, method  300  may be performed using arrangement  200  of  FIG. 2  to determine the linearity of DUT  50 . For clarity of explanation, method  300  will be described with respect to arrangement  200 , although it will be understood that the method may be performed using other arrangements, including for example the arrangements shown in  FIGS. 5 and 7  which are described below. 
     In a step  310 , a first periodic signal (e.g., the first sinusoidal signal S 1 ) is generated, and a second periodic signal (e.g., the second sinusoidal signal S 2 ) is generated, where second sinusoidal signal has a phase and/or frequency difference or offset with respect to first sinusoidal signal S 1 . In some embodiments as described below, processor  240  supplies one or more control signals to one or both of first signal generator  210  and second signal generator  215  as illustrated in  FIG. 2  to control a frequency and/or phase of the first periodic signal S 1  and/or the second periodic signal S 2 . 
     In a step  320 , the first sinusoidal signal S 1  and the second sinusoidal signal S 2  are combined by signal combiner  230  to produce a combined signal S 3 . 
     In a step  330 , the combined signal S 3  is applied to DUT  50 . In response, DUT  50  generates an output signal S 4 . 
     In a step  340 , the linearity of DUT  50  is determined from output signal S 4  based on the phase or frequency difference between the first and second sinusoidal signals S 1  and S 2 . In particular, in some embodiments the second sinusoidal signal S 2  has the same frequency as the first sinusoidal signal S 1 , but has a phase difference with respect to the first sinusoidal signal S 1 , and that phase difference is varied to determine the linearity of DUT  50 . In other embodiments, the second sinusoidal signal S 2  has a frequency difference with respect to the first sinusoidal signal S 1  that produces a predetermined time-varying magnitude for the combined signal S 3  that depends on that frequency difference, and the output signal S 4  of DUT  50  is compared to the expected response of an ideal DUT when receiving the combined signal S 3  having the predetermined time-varying magnitude, in order to determine the linearity of DUT  50 . 
     Detailed explanations of example embodiments of method  300  will now be provided. 
     In a first example embodiment of method  300  of determining the linearity of DUT  50 , the second periodic signal S 2  has a phase difference with respect to the first periodic signal S 1 , and this phase difference is varied to determine the linearity of DUT  50 . A more detailed description of such an example embodiment is now provided. 
     In general, one can express the first and second periodic signals S 1  and S 2  having the same frequency, but having a phase difference with respect to each other, as:
 
 S 1= A 1*cos(ω* t +φ), and  (1)
 
 S 2= A 2*cos(ω* t +θ),  (2)
 
where ω=2πf, f=frequency, t=time, and φ &amp; θ and A 1  &amp; A 2  are scalar constants, and A 1  and A 2  represent the peak amplitudes of S 1  and S 2 , respectively.
 
     The combined signal, S 3 , is the sum of S 1  and S 2 , with an additional gain factor and phase offset due to the loss and delay of signal combiner  230 , and can be expressed as:
 
 S 3= B 1*cos(ω* t +φ′)+ B 2*cos(ω* t +θ′),  (3)
 
where φ′ and θ′ are scalar constants representing the phase values φ and θ, respectively, phase shifted by corresponding phase amounts by signal combiner  230 , and B 1  and B 2  are scalar constants representing the peak amplitudes A 1  and A 2 , respectively, adjusted by corresponding amplitude gains/losses from signal combiner  230 . Thus it is seen that the combined signal S 3  includes a component of the first periodic signal S 1 , having a peak amplitude B 1  and a phase value φ′, and further includes a component of the second periodic signal S 2  having a peak amplitude B 2  and a phase value θ′.
 
     Although in general B 1  and B 2  can have any realizable values, in a beneficial embodiment, A 1  and A 2  are selected such that the peak amplitude of the component of the first periodic signal present in the combined signal, B 1  equals the peak amplitude of the component of the second periodic signal present in the combined signal, B 2 . In that case: 
     
       
         
           
             
               
                 
                   
                     
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     When DUT  50  is a typical detector, it will detect the envelope of the combined signal S 3  which is the absolute value or magnitude of B 3 . Thus, as shown in equation (4), by varying the phase difference Δθ between the phase φ′ of the component of the first periodic signal S 1  present in the combined signal S 3 , and the phase θ′ of the component of the second periodic signal S 2  present in the combined signal S 3 , one can vary the value of B 3 . Since the power level of the combined signal S 3  is proportional to the square of the magnitude or absolute value of B 3 , then for a constant impedance system the ratio of the power incident upon DUT  50  for two phase difference values, Δθ 1  and Δθ 2 , is: 
                                S   ⁢           ⁢     3   2            2              S   ⁢           ⁢     3   1            2       =              cos   ⁡     (       Δϑ   2     2     )            2              cos   ⁡     (       Δϑ   1     2     )            2         ,           (   8   )               
where S 3   1  is the combined signal for the phase difference value Δθ 1 , and S 3   2  is the combined signal for the phase difference value Δθ 2 .
 
     It should be noted that the amplitude B 1  (=B 2 ) is constant and drops out of equation (8), so that the measurements do not depend on the amplitudes of the first and second sinusoidal signals S 1  and S 2 . It should also be noted that the phase difference Δθ depends on the phase values φ and θ of the first and second sinusoidal signals S 1  and S 2 , so—for example—the phase difference Δθ can be changed from Δθ 1  to Δθ 2  by adjusting the phase value φ or θ of the corresponding one of the first and second signal generators  210  and  215  while keeping the other phase value constant. 
     In general, to measure the linearity of DUT  50 , in one example embodiment the phase value θ of second signal generator  215  is kept constant while the phase value φ of first signal generator  210  is varied to cause the phase difference Δθ have a plurality of different values over a desired phase difference range, and the output signal S 4  of DUT  50  is analyzed by processor  240  to compare the measured response of DUT  50  to the theoretical response for an ideal linear device as provided in equation (8). 
       FIG. 4  illustrates one example embodiment of an output response of DUT  50  in the arrangement of  FIG. 2  as a function of the phase difference Δθ between the component of the first periodic signal S 1  present in combined signal S 3 , and the component of the second periodic signal S 2  present in combined signal S 3 , when DUT  50  is an ideal device. In  FIG. 4 , the solid line  410  represents the peak amplitude B 1  (=B 2 , as described above), and the dotted curve  420  represents the absolute value or magnitude of the peak amplitude B 3  (normalized with respect to B 1 ) of the combined signal S 3  at the input of DUT  50  as a function of the phase difference Δθ between the component of the first periodic signal S 1  present in the combined signal S 3 , and the component of the second periodic signal S 2  present in S 3 . As can be seen in  FIG. 4 , dotted curve  420  represents the absolute value or magnitude of a cosine function, and particularly: 
     
       
         
           
             
               
                 
                   
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     The method of measuring the linearity of DUT  50  described above assumes that the amplitude B 2 =B 1 , and that the phase difference Δθ is either known or can be determined for each measurement point at which the output signal S 4  is analyzed. Several different methods for setting the amplitude B 2 =B 1 , and for determining the phase difference Δθ may be employed. 
     In one embodiment, the phase difference Δθ is controlled by controlling the phase(s) of one or both of first and second signal generators  210  and  215 . In an example embodiment as shown in  FIG. 2 , processor  240  may provide one or more control signals to one or both of first and second signal generators  210  and  215  for controlling the phase(s) of the first and/or second sinusoidal signal(s) S 1  and/or S 2 . In particular, processor  240  may vary the phase difference Δθ in known increments and analyze the output signal S 4  of DUT  50  as the phase difference is varied to find the maximum point  422 , the minimum point  424 , and/or the single power point  450  (i.e., the point where B 3   2 =B 1   2 ) as shown in  FIG. 4 . Further known changes in the phase difference Δθ may then be used to change the power of the combined signal S 3  supplied to DUT  50  according to equation (5). The linearity of DUT  50  may be determined by comparing the measured change from the output signal S 4  of DUT  50  to the change in the absolute value or magnitude of the peak amplitude of S 3  at different phase differences Δθ according to equation (8) and/or  FIG. 4 . 
     In one embodiment, the amplitudes B 1  and B 2  can be set equal to each other as follows. First, the sinusoidal signal S 2  from second signal generator  215  is turned off (for example by a switch), as a result of which the amplitude of the first sinusoidal signal applied to signal combiner  230  is set to zero, and the power level B 3   2 =B 1   2 . Then the combined signal S 3  having the power level B 3   2  is applied to DUT  50  while the peak amplitude A 1  is adjusted until the output signal S 4  indicates a first value for B 1 . Next, the first sinusoidal signal S 1  from first signal generator  210  is turned off (for example by a switch), as a result of which the amplitude of the second sinusoidal signal applied to signal combiner  230  is set to zero, and the power level B 3   2 =B 2   2 , and then A 2  is adjusted until the output signal S 4  indicates the same first value for B 1  as was determined before. At that point, B 2  will be equal to B 1 . It should be noted that by requiring only that B 1  and B 2  be fixed at the same value, DUT  50  need not be linear, as long as its output signal S 4  has a unique value or level for each input power level B 3   2 . It should also be noted that the power level where B 2 =B 1  also provides markers for the phase difference response, as discussed above with respect to  FIG. 4 . 
     In another embodiment, the peak amplitudes B 1  and B 2  and the phase difference Δθ may be determined using separate detectors, as illustrated with respect to  FIG. 5 . 
       FIG. 5  illustrates another example embodiment of an arrangement  500  for measuring the linearity of DUT  50  using two periodic signals. The arrangement  500  is identical to the arrangement  200  described above, except for the addition of: a first directional coupler  522  in a signal path between first signal generator  210  and signal combiner  230 ; a first detector  523  connected to a coupling output port of first directional coupler  522  and configured to output to processor  240  a first detected signal; a second directional coupler  524  in a signal path between second signal generator  215  and signal combiner  230 ; and a second detector  525  connected to a coupling output port of second directional coupler  524  and configured to output to processor  240  a second detected signal. 
     As a beneficial feature, first and second directional couplers  522  and  524  are time invariant linear directional couplers and first and second detectors  523  and  525  are time invariant linear detectors. Because first and second directional couplers  522  and  524  and first and second detectors  523  and  525  are linear and time invariant, the difference between the detected phases of a first coupled signal SR 1  from first directional coupler  522  and a second coupled signal SR 2  from second directional coupler  524  is equal to Δθ plus a constant offset. The same detectors (first and second detectors  523  and  525 ) can also be used to measure the amplitudes of first sinusoidal signal S 1  (i.e., A 1 ), and second sinusoidal signal S 2  (i.e., A 2 ). 
     Using the process described above, the output signal levels of first and second signal generators  210  and  215  are set so that B 1  and B 2  are equal. Then, the phase difference Δθ between the first and second sinusoidal signals S 1  and S 2  is varied until the power level detected by DUT  50  is either: 
     (I) At a maximum point  422  as shown in  FIG. 4 ; in this case, Δθ is 0; 
     (II) At a minimum point  424  as shown in  FIG. 4 ; in this case, Δθ is π radians. 
     (III) At the same level as the single source power condition. In this case, B 3 =B 1  and therefore cos(Δθ)=0.5, as shown at the single power points  450  of the solid line  410  and the dotted curve  420  in  FIG. 4 . 
     Option (I) is problematic due to the tiny variation in power level around the maximum point  422  leading to phase uncertainty. Option (II) is problematic due to the exaggerated effects of noise at the minimum point  424 . Meanwhile, Option (III) does not depend upon the linearity of first and second detectors  523  and  525 , is less susceptible to noise than Option (II), and is at a point in the waveform at which the power level changes quickly with phase, so that the phase uncertainty is minimized. Option (III) can be produced at two different values of Δθ, but these two values are easily distinguishable. 
     At one of the above power levels for which the phase difference Δθ of the combined signal S 3  is known, the first and second detectors  523  and  525  measure the difference in the phase between first and second coupled signals SR 1  and SR 2 , and the difference between this measurement and the known phase difference Δθ of the combined signal S 3  is recorded and used as an offset to determine the phase difference Δθ in subsequent measurements. 
     Also, it should be noted that the signal amplitude measured by reference detector R 1  is proportional to B 1 , and the signal amplitude measured by reference detector R 2  is proportional to B 2 . This fact can be used to maintain a constant value for the peak amplitudes A 1  and A 2  of the first and second sinusoidal signals S 1  and S 2  output by first and second signal generators  210  and  215 . 
     In a second example embodiment of method  300  of determining the linearity of DUT  50 , the second periodic signal S 2  has a frequency difference with respect to the first periodic signal S 1 , and the linearity of DUT  50  is determined based on this frequency difference. A more detailed description of such an example embodiment is now provided. 
     In general, one can express the first and second periodic signals S 1  and S 2  having a constant frequency difference and a phase difference with respect to each other, as:
 
 S 1= A 1*cos(ω* t +φ), and  (10)
 
 S 2= A 2*cos [(ω+δ)* t+θ],   (11)
 
where ω=2πf, f=frequency, t=time, δ is the constant frequency difference between the second periodic signal S 2  and the first periodic signal S 1 , and φ and θ and A 1  and A 2  are scalar constants, as before.
 
     The combined signal, S 3 , is the sum of the first periodic signal S 1  and the second periodic signal S 2 , with an additional gain factor and phase offset due to the loss and delay of signal combiner  230 , and can be expressed as:
 
 S 3= B 1*cos(ω* t +φ′)+ B 2*cos [(ω+δ)* t +θ′],  (12)
 
where φ′ and θ′ are scalar constants representing the phase values φ and θ, respectively, phase shifted by corresponding phase amounts by signal combiner  230 , and B 1  and B 2  are scalar constants representing peak amplitudes A 1  and A 2  adjusted by corresponding amplitude gains/losses by signal combiner  230 , as before. Thus it is seen that the combined signal S 3  includes a component of the first periodic signal S 1  having a peak amplitude B 1  and a phase value φ′, and further includes a component of the second periodic signal S 2  having a peak amplitude B 2  and a phase value θ′.
 
     As before, in general B 1  and B 2  can have any realizable values, but in a beneficial embodiment, A 1  and A 2  are selected such that B 1 =B 2 . In that case: 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       3 
                     
                     = 
                     
                       
                         B 
                         3 
                       
                       * 
                       
                         cos 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   ω 
                                   + 
                                   
                                     δ 
                                     2 
                                   
                                 
                                 ) 
                               
                               * 
                               t 
                             
                             + 
                             Φ 
                           
                           ] 
                         
                       
                     
                   
                   , 
                   
                     where 
                     ⁢ 
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     = 
                     
                       2 
                       * 
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                       * 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             Θ 
                             - 
                             
                               
                                 δ 
                                 * 
                                 t 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     Θ 
                     = 
                     
                       
                         
                           ϕ 
                           ′ 
                         
                         - 
                         
                           θ 
                           ′ 
                         
                       
                       2 
                     
                   
                   ; 
                   and 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   Φ 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             ϕ 
                             ′ 
                           
                           + 
                           
                             θ 
                             ′ 
                           
                         
                         ) 
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Again, when DUT  50  is a typical detector, it will detect the envelope of the combined signal S 3  which is the absolute value or magnitude of B 3 . Thus, as shown in equation (14), DUT  50  detects the absolute value of 
             2   *   B   ⁢           ⁢   1   *     cos   ⁡     (     Θ   -       δ   *   t     2       )             
which is a periodic signal as a function of time with a frequency of δ/2 radians per second.
 
       FIG. 6  illustrates an example embodiment of an output response of DUT  50  as a function of time in the arrangement of  FIG. 2  when DUT  50  is an ideal device and there is a fixed frequency difference between the second periodic signal S 2  and the first periodic signal S 1 . In particular,  FIG. 6  plots the absolute value or magnitude of 
             2   *   B   ⁢           ⁢   1   *     cos   ⁡     (     Θ   -       δ   *   t     2       )             
(i.e., an absolute value or magnitude of a cosine function of the frequency difference δ multiplied by time t) as a function of time, with δ=2*π*6, Θ≈20°, and B 1 =0.1. The circled intercept points in  FIG. 6  are key defined points that may be used to extract the actual values of the frequency difference δ and the phase difference value Θ from the output signal S 4  of DUT  50 .
 
     In some embodiments, DUT  50  may make multiple measurements of the absolute value or magnitude of B 3  of the combined signal S 3 , and processor  240  may average the comparison between the measured absolute value or magnitude of B 3  and the ideal response of  FIG. 6  over several periods of the cosine function. Averaging may be employed to reduce jitter and noise. 
     In some embodiments, processor  240  analyzes the output signal S 4  of DUT at periodic intervals. In that case, the frequency difference δ may be selected such that many periodic measurements of the absolute value or magnitude of B 3  are made by DUT  50  for each period of the combined signal S 3 . 
     The phase difference Θ is determined from observation of the periodic response of the output signal S 4  of DUT  50 . In particular, by observing the time period between the circled points in  FIG. 6  at which |S 3 |=B 1 , the values of Θ and δ can be determined as shown in  FIG. 6 . It should be noted that this determination is not affected by the linearity of DUT  50 , since all determinations are made with the same power applied to DUT  50 . It should also be noted that accuracy of the frequency settings and measurement rate settings are not critical. 
     The frequency difference δ may also be determined from a priori knowledge of the frequency difference or offset between the first and second sinusoidal signals S 1  and S 2  output by first and second signal generators  215  and  220  based on the settings of first and second signal generators  215  and  220 , which may be controlled by processor  240  as shown for example in  FIGS. 2 and 5 . 
     The phase difference value Θ and the frequency difference δ may also be determined by modifying their values to minimize the difference between the formula |B 3 |=|2*B 1 *cos(δ*t/2−Θ| and the observed response of the output signal S 4  of DUT  50 , and also by observing the number of sample points between crossings of B 1 . This minimization process does not impact the end results since the observed data is preserved and only the systematic errors of instrument settings are removed. 
     If filters are used in DUT  50 , it is important to keep ω and ω+δ within the flat region of the filter&#39;s passband to minimize linearity errors due to the filter. 
     In some embodiments of the frequency difference method of determining the linearity of DUT  50 , a change in the output signal S 4  at two different times may be compared against the ideal response as found in equation: 
                                S   ⁢           ⁢     3   2            2              S   ⁢           ⁢     3   1            2       =              cos   ⁡     (         δ   *     t   2       2     -   Θ     )            2              cos   ⁡     (         δ   *     t   1       2     -   Θ     )            2         ,           (   17   )               
where S 3   1  is the combined signal at time t 1 , and S 3   2  is the combined signal at the time t 2 .
 
     As illustrated in  FIGS. 4 and 6 , the cosine function has a very steep slope as the phase change approaches 180 degrees. This increases the sensitivity of the output signal S 4  of DUT  50  to small errors in time or phase. To reduce this sensitivity, in some embodiments the measurements are limited to a range of 145° of phase change in the cosine function. This is equivalent to 12 dB of input power level change. 
     Toward that end,  FIG. 7  illustrates yet another example embodiment of an arrangement  700  for measuring the linearity of DUT  50  using two periodic signals. The arrangement  700  is identical to the arrangement  500  described above, except for the addition of a variable attenuator  720 . That is, in order to measure the linearity of DUT  50  over a wider range of changes in power level, variable attenuator  720  is added to the arrangement, as illustrated by  FIG. 7 , to increase the measurement dynamic range. In the arrangement  700 , variable attenuator  720  is stepped through increasing attenuation values, for example in 10 dB increments, and linearity measurements are made as described above for the attenuated combined signal at each attenuation value. 
       FIG. 8  illustrates the absolute value or magnitude of B 3  as a function of time for the combined signal S 3  that is provided to DUT  50  for various attenuation values of variable attenuator  720  in an example embodiment of arrangement  700 .  FIG. 8  also illustrates an ideal output response of DUT  50  as a function of time in the example embodiment of arrangement  700  for various attenuation values provided by variable attenuator  720  when the first and second sinusoidal signals S 1  and S 2  have a fixed frequency difference between them. Attenuation values of 0 dB, 10 dB and 20 dB are shown in the plots  810 ,  820  and  830 , respectively. This technique allows the combined signal S 3  to cover a very wide dynamic range while avoiding the high sensitivity region of the cosine function around 180 degrees, as seen for example in  FIGS. 4 and 6 . Since 12 dB of power change is covered for each 10 dB step attenuation value, overlapping power level points are provided so that processor  240  can stitch together the plots  810 ,  820  and  830  to produce a response covering a wide dynamic range. Since only ratios are employed, the absolute accuracy of variable attenuator  720  is irrelevant. The amplitudes of the first sinusoidal signal S 1  and the second sinusoidal signal S 2  are kept constant throughout the linearity measurements with the attenuated combined signal. 
     In particular, processor  240  determines a first portion of the time varying characteristic of the output signal S 4  of DUT  50  over a specified range of angles less than 180 with the combined signal S 3  applied to the DUT  50  at a first attenuation setting (e.g., 0 dB). Then the attenuation of variable attenuator  720  is adjusted to provide multiple power settings. For example, the attenuation may be adjusted in steps of 10 db, for example by processor  240 . Processor  240  generates a second portion of the time varying characteristic of the output signal S 4  of the DUT  50  with the attenuated combined signal S 3  applied to DUT. This process may be repeated again for additional attenuation values (e.g., in 10 dB steps) as desired to obtain additional portions of the time varying characteristic of the output signal S 4  of the DUT  50 . Finally, processor  240  stitches together the first portion of the time varying characteristic of the output signal S 4  and the second portion of the time varying characteristic of the output signal S 4  (and any additional portions that were acquired) (and any additional portions of the time varying characteristic of the output signal S 4  produced at additional attenuation settings) to produce the time varying characteristic of the output signal. 
       FIG. 9  shows a plot  900  that illustrates and compares examples of the mean error and the repeatability &amp; reproducibility variation (R&amp;R) of linearity measurements made using an example embodiment of an arrangement as illustrated  FIG. 7 , and an example arrangement as illustrated in  FIG. 1B . In particular,  FIG. 9  plots mean error, R&amp;R, and (mean error+3 R&amp;R) values as a function of input power level at a DUT of linearity measurements performed using an example embodiment of an arrangement as illustrated  FIG. 7 , and an example arrangement as illustrated in  FIG. 1B . As can be seen in  FIG. 9 , the mean error of the linearity measurements made using the example embodiment of the arrangement as illustrated  FIG. 7  is less than the mean error of the linearity measurements made using the example arrangement as illustrated in  FIG. 1B , and the R&amp;R of the linearity measurements made using the example embodiment of an arrangement as illustrated  FIG. 7  is less than the R&amp;R of the linearity measurements made using the example arrangement as illustrated in  FIG. 1B . 
     In embodiments described above, methods have been described for determining device linearity by employing systems with first and second signal generators  210  and  215 , which can provide an output signal with a power change that can be precisely determined by the phase difference or frequency difference between the first and second signal generators  210  and  215 . However such systems may be employed in a variety of other contexts besides measuring device linearity. One example use of such a system is to determine whether or not one or more particular RF signals are present or absent at a particular location, for example to address the so-called “white-space” problem with some smart radio systems that search for TV or radio channels that are unoccupied so that these channels may be used for wireless communications by the smart radio systems. In some of these applications, systems such as arrangements  200 ,  500 , and  700  of  FIGS. 2, 5 and 7  may be employed, with a receiver configured to measure the phase difference or the frequency difference between the first and second signal generators  210  and  215  substituted in place of the device-under-test  50 . 
     While example embodiments are disclosed herein, one of ordinary skill in the art appreciates that many variations that are in accordance with the present teachings are possible and remain within the scope of the appended claims. The invention therefore is not to be restricted except within the scope of the appended claims.