Patent Application: US-48486509-A

Abstract:
methods for analyzing circuit distortion based on contributions from separate circuit elements are presented . local approximations that do not require high - order derivatives of device models are developed near an operating point for calculating distortion summaries including compression summaries and second - order intermodulation distortion summaries .

Description:
fig1 shows a system 102 ( e . g ., a circuit model ) that receives an rf input 104 with an input frequency ω rf and produces an output 106 where the input frequency has been modulated by the system ( e . g ., shifted by a multiple of the lo ( local oscillator ) frequency ω lo . under non - ideal conditions , the output 106 includes distortions due to the nonlinear effects in the system 102 . as described below , the present invention enables quantitative characterizations of how component devices contribute to distortion in the output signal 106 . specific embodiments include calculations for compression distortion and second - order intermodulation products ( im2 ). this application is related to calculating intermodulation products and intercept points for circuit distortion analysis ( application ser . no . 11 / 303 , 049 , filed dec . 14 , 2005 ), which is incorporated herein by reference in its entirety . the circuit 102 in fig1 has a periodic time - dependent operating point v 0 ( t ) at the lo frequency ω lo more generally , systems with dc ( direct current ) operating points such as an lna ( low noise amplifier ) can be considered as special cases with ω lo = 0 . when the rf input signal 104 is applied , the output signal 106 at ω rf + nω lo is a function of the input magnitude v rf and given as : v out ( ω rf + nω lo )= c 1 v rf + c 3 v rf 3 + ( 1 ) the first term on the right - hand side of eq . ( 1 ) is the linear response to rf input , and the coefficient c 1 can be obtained by small signal calculation . other terms model distortion due to third and higher order terms in the nonlinear response . the compression distortion can be measured in db ( decibels ) as while it &# 39 ; s desirable to know the contribution of each individual device to total compression , it is mathematically intractable to make such a separation for third or higher order nonlinearities . in order to quantify the effect of a specific device on the nonlinear response at the output signal , we assume the rest of the circuit responds to the rf signal linearly and compute the distortion in the output caused solely by nonlinear response at this device . with this approach , although the circuit is hypothetical , the result provides useful information about how sensitive distortion is to the device . the effects of the device on the nonlinear response and the transfer function from device nodes to output nodes are reflected in the calculation . notably , since the operating point is time - dependent , all the other devices are still nonlinear in the calculations . only their response to rf signal is linearized . in particular , for mixer cases , transistors in lo block still function nonlinearly to generate the clock signal . however , their contribution to the output distortion summary is expected to be very small because the rf signal 104 usually does not disturb the lo part of the circuit . both linear and nonlinear responses in lo block are typically almost zero . we formulate compression distortion summary calculations by writing circuit equation as ⅆ ⅆ t ⁢ q ⁡ ( υ ⁡ ( t ) ) + i ⁡ ( υ ⁡ ( t ) ) = b ⁡ ( t ) + s ⁡ ( t ) , ( 3 ) where b is the operating point source such as lo at ω lo and s is rf input signal at ω rf [ 13 ]. for simplicity , we rewrite eq . ( 3 ) as assume v 0 ( t ) is the steady - state ( or dc ) solution at zero rf input so that v 0 satisfies equation where v is the circuit response to rf signal s . the linear operator l in the first term is defined as l · v = ⅆ ⅆ t ⁢ ( ∂ ∂ υ ⁢ q ⁡ ( v 0 ) · v ) + ∂ ∂ υ ⁢ i ⁡ ( v 0 ) · v ( 7 ) and l · v is just the linearized lhs of eq . ( 3 ). the second term f nl represents the sum of nonlinear terms in all the devices and is defined as f nl ( v 0 , v )= f ( v 0 + v )− f ( v 0 )− l · v . ( 8 ) that is , linearizing about the dc or operating point v 0 determines linear and nonlinear offsets ( or offset functions ), l and f nl , for analyzing local perturbations of the output . note that f nl is the sum of nonlinear contribution from each device . to calculate distortion caused solely by device d , we omit f nl in all other devices except in d . eq . ( 6 ) then becomes compression due to d is computed with v and v ( 1 ) at frequency ω rf + nω lo ( n is an integer ): db = 10 · log ⁡ [ v ⁡ ( ω rf + n ⁢ ⁢ ω lo ) v ( 1 ) ⁡ ( ω rf + n ⁢ ⁢ ω lo ) ] ( 11 ) in most analog designs , the circuit functions in a nearly linear region . the nonlinear term f nl ( d ) in eq . ( 10 ) is small compared to the linear term l · v and v is not far away from v ( 1 ) . therefore , we can treat f nl ( d ) as perturbation and apply born approximation to solve v iteratively u ( n ) = v ( 1 ) − l − 1 · f nl ( d ) ( u ( n - 1 ) ), ( 12 ) where u ( n ) is the approximated v accurate up to nth order of v rf for n ≧ 2 and u ( 1 ) is the same as v ( 1 ) . eq . ( 12 ) is equivalent to ac or periodic ac analysis with f nl ( d ) ( v ( n - 1 ) ) being the small signal . as shown in eq . ( 8 ), evaluation of f nl ( d ) takes a function evaluation of f at device d and its first derivative . no modification is needed in the device model . in compression distortion summary , we solve v up to 3rd order perturbation and use u ( 3 ) as v in eq . ( 11 ). to simplify eq . ( 12 ) we define rf harmonic of a multi - tone function ƒ ( t ) as f ω ⁡ ( t ) = ∑ m ⁢ ⁢ f ⁡ ( ω + m ⁢ ⁢ ω lo ) ⁢ ⅇ j ⁡ ( ω + m ⁢ ⁢ ω lo ) ⁢ t ( 13 ) where ω is the frequency a particular rf harmonic and ƒ ( ωmω lo ) is the fourier transform of ƒ ( t ). eq . ( 12 ) can be written in rf harmonic form as u ω ( n ) = v ω ( 1 ) − l − 1 · f nl , ω ( d ) ( u ( n - 1 ) ) ( 14 ) rf harmonic f nl , ω ( d ) of f nl ( d ) can be obtained by dft ( discrete fourier transform ). for compression distortion , we solve ( 14 ) at the first rf harmonic ω = ω rf to 3rd order . u ω rf ( 3 ) = v ω rf ( 1 ) − l − 1 · f nl , ω rf ( d ) ( u ( 2 ) ) ( 15 ) notice that as defined in ( 8 ) f nl ( v ) is the nonlinear part of function f ( v ) and its lowest order polynomial is v 2 . third order terms in u ω rf ( 3 ) equation arise from 2nd and 3rd in ( and polynomials in f nl which involve only rf harmonics at ω rf and − ω rf in v ( 1 ) and 2ω rf and 0 in u ( 2 ) . u ω rf ( 3 ) = v ω rf ( 1 ) − l − 1 · f nl , ω rf ( d ) ( v ω rf ( 1 ) + v − ω rf ( 1 ) + u 2ω rf ( 2 ) + u 0 ( 2 ) ) ( 16 ) eq . ( 16 ) shows that to solve u ω rf ( 3 ) at third order we only need to solve u 2ω rf ( 2 ) and u 0 ( 2 ) in second - order born approximation , for which we have u 2ω rf ( 2 ) =− l − 1 · f nl , 2ω rf ( d ) ( v ( 1 ) ) ( 17 ) u 0 ( 2 ) =− l − 1 · f nl , 0 ( d ) ( v ( 1 ) ). ( 18 ) other rf harmonics in u ( 2 ) contribute to higher order terms , and therefore can be ignored in 3rd order perturbation . ( note that in eq . ( 16 ) v ω rf ( 1 ) and v − ω rf ( 1 ) are complex conjugates so that only one solution must be obtained .) fig2 shows a method for calculating a compression distortion summary 202 consistent with the above discussion . first , a dc or operating point is determined 204 ( e . g ., approximated in the time domain [ 13 ] or the frequency domain [ 16 ]). next eq . ( 9 ) is solved for the first - order linear response v ( 1 ) at ω rf 206 . note that eq . ( 9 ) can be solved in the time domain [ 13 ] or alternatively in the frequency domain [ 16 ] by standard methods . the process then continues for one or more devices ( e . g , component elements d of the system 102 ). first the nonlinear terms are calculated at frequencies 2ω rf and zero 208 . next , second - order solutions are calculated for these frequencies 210 . next the nonlinearities in eq . ( 16 ) are evaluated at ω rf 212 , and the equation is solved for the third - order solution u ( 3 ) 214 . this process can be continued for all devices in the system 102 or a limited subset of component devices . simulation results corresponding to an implementation of this method are shown in fig3 and 4 . fig3 shows a compression distortion summary ( in db ) for mixer ne600 ( philips semiconductors ) at − 15 dbm at node pif . compressions due to all devices and due to each individual device are listed . results based on time - domain shooting and harmonic - balance implementations are given and they are consistent with each other . fig4 shows similar results ( based on time - domain shooting ) for a compression distortion summary ( in db ) of a pa ( power amplifier ) at 0 . 1v at node rfout . in direct conversion mixer design , second - order intermodulation ( im2 ) between two rf input signals is used to measure circuit distortion . as a second order effect , im2 does not involve crossing nonlinear terms between devices . it is simply the sum of second order terms in each individual device . this can be demonstrated by second order born approximation for im2 at frequency ω 1 - ω 2 where ω 1 and ω 2 are frequencies of the two rf input signals . as shown in eq . ( 19 ), evaluation of f nl ( d ) ( v ( 1 ) ) takes only linear response v ( 1 ) at second order perturbation . hence , f nl ( d ) ( v ( 1 ) ) is solely due to nonlinearity in device d and no other device is involved in it . the same argument also holds in volterra series based perturbation . we apply the adjoint analysis to compute contribution of each device to im2 at the relevant output nodes . using transfer function x ω 1 - ω 2 , eq . ( 19 ) can be rewritten as inner products of x ω 1 - ω 2 and f nl , ω 1 - ω 2 ( d ) ( v ( 1 ) ): v out ⁡ ( ω 1 - ω 2 ) = ∑ d ⁢ ⁢ ( x ω 1 - ω 2 , - f nl , ω 1 - ω 2 ( d ) ⁡ ( v ( 1 ) ) ) ( 20 ) where v out ( ω 1 - ω 2 ) is im2 measured at the output nodes at frequency ω 1 - ω 2 . note that in this context the transfer function x ω 1 - ω 2 is a projection of the solution operator ( e . g ., green &# 39 ; s function ) onto a combination of one or more output nodes where v out ( ω 1 - ω 2 ) is being measured ( e . g . a difference of voltages at two output nodes ). this approach reduces the computational cost since x ω 1 - ω 2 is independent of f nl . once x ω 1 - ω 2 is solved by the linear adjoint analysis , then instead of solving eq . ( 19 ) repeatedly , the contribution of element d can be computed by simply calculating the inner product of x ω 1 - ω 2 and f nl , ω 1 - ω 2 ( d ) ( v ( 1 ) ). alternatively , the complete solution of eq . ( 19 ) can be calculated before making a projection onto the relevant output nodes . however , the solution of eq . ( 19 ) is generally only required at a scalar combination of output nodes where the distortion is being measured . therefore , calculating the im2 distortion summary according to eq . ( 20 ) is generally preferred . fig5 shows a method for calculating an im2 distortion summary 502 consistent with the above discussion . first , a dc or operating point is determined 504 ( e . g ., approximated in the time domain [ 13 ] or the frequency domain [ 16 ]). next eq . ( 9 ) is solved for the first - order linear response v ( 1 ) at ω 1 and ω 2 506 . as noted above , eq . ( 9 ) can be solved in the time domain [ 13 ] or alternatively in the frequency domain [ 16 ] by standard methods . next the transfer function x ω 1 - ω 2 is characterized as a projection of the solution operator onto the combination of outputs where the distortion is being measured 508 . note that this transfer function x ω 1 - ω 2 can be separately calculated or calculated implicitly when determining the inner products as discussed below 512 . the process then continues for one or more devices ( e . g , component elements d of the system 102 ). the nonlinearities in eq . ( 20 ) are evaluated at ω 1 - ω 2 510 , and the inner products are evaluated 512 for characterizing the distortion v out ( ω 1 - ω 2 ). this process can be continued for all devices in the system 102 or a limited subset of component devices . simulation results corresponding to an implementation of this method are shown in fig6 . fig6 shows im2 distortion summary results in v for a direct conversion mixer . total im2 at output nodes and contribution of each individual device are listed . results from time - domain shooting - based and harmonic balance - based implementations are given and they are consistent to each other . additional embodiments relate to an apparatus for carrying out any one of the above - described methods , where the apparatus may include a computer for executing computer instructions related to the method . in this context the computer may be a general - purpose computer including , for example , a processor , memory , storage , and input / output devices ( e . g ., monitor , keyboard , disk drive , internet connection , etc .). however , the computer may include circuitry or other specialized hardware for carrying out some or all aspects of the method . in some operational settings , the apparatus may be configured as a system that includes one or more units , each of which is configured to carry out some aspects of the method either in software , in hardware or in some combination thereof . at least some values based on the computed distortion summaries can be saved , either in memory ( e . g , ram ( random access memory )) or permanent storage ( e . g ., a hard - disk system ) for later use . for example , values for the left - hand sides of eqs . ( 11 ) and ( 20 ) can be used directly as distortion summaries . alternatively , values for components of the right - hand sides of these equations can also be saved for later use . and similarly related values ( e . g ., averages ) can also be saved depending on the requirements of the operational setting . additional embodiments also relate to a computer - readable medium that stores ( e . g ., tangibly embodies ) a computer program for carrying out any one of the above - described methods by means of a computer . the computer program may be written , for example , in a general - purpose programming language ( e . g ., c , c ++) or some specialized application - specific language . the computer program may be stored as an encoded file in some useful format ( e . g ., binary , ascii ). although only certain exemplary embodiments of this invention have been described in detail above , those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings and advantages of this invention . for example , aspects of embodiments disclosed above can be combined in other combinations to form additional embodiments . accordingly , all such modifications are intended to be included within the scope of this invention . j . bussgang , l . ehrman , and j . graham . analysis of nonlinear systems with multiple inputs . proc . ieee , 62 : 1088 - 1119 , august 1974 . 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