Relation Between Multitone and Two-Tone Test Results

2.4 Multitone or Continuous Spectra Characterization Tests

2.4.6 Relation Between Multitone and Two-Tone Test Results

Before closing the analysis of multitone tests, it is interesting to relate their figures of merit with the ones previously derived for the two-tone stimulus. In fact, since two-tone tests still represent the most widely used nonlinear distortion characteriza- tion method, there are many situations where a certain device comes specified with the standardIP3, and we need to estimate its impact in a real multitone or continu- ous spectrum application. Or, conversely, we may want to specify such a device in terms of itsIP3, orIMR,from the knowledge of its admissible distortion under the actual multitone or continuous spectrum environment.

Unfortunately, this task is mathematically very involved, being intractable for all but a very few special cases. From these, the one providing most useful results,

Figure 2.22 Cochannel and adjacent-channel border definition for accurate distortion measure- ments under excitations of finite slope roll-off.

from a practical point of view, assumes the system is a third-order memoryless nonlinearity, excited byQevenly spaced but uncorrelated tones of equal amplitude.

Third-order mixing products of these tones produce spectral regrowth compo- nents whose new frequency positions are given by␻r=␻q1+␻q2−␻q3, in which q1≠q2≠q3(products from now on named as Type A), q1=q2≠q3(Type B);

and also generate products falling on the same positions of the input, in which q1≠q2=q3(Type C) and q1=q2=q3(Type D).

Now, to calculate the magnitude of adjacent or cochannel distortion compo- nents shown in Figure 2.23, we need to calculate the number of different mixing products appearing in each frequency position.

For that, we begin by first determining the number of different ways the set of three input frequencies, ␻r=␻q1+␻q2−␻q3, can be grouped to produce a certain mixing component. That number is the multinomial coefficient of the mixing product [given by (1.23)], and values 6 for Type A products, 3 for type B, 6 for Type C, and finally 3 for type D.

The second step consists of calculating the number of possible combinations of input tones that produce mixing products at the same frequency position, ␻r. That derivation was described in [11] and used straightforward, although quite laborious, combinatory calculus. Its results for the number of mixing products located at the adjacent-channel,Q+1≤r≤2Q−1, in Figure 2.23(a) were

Figure 2.23 Identification of the output spectrum’s frequency positions corresponding to (a) adjacent-channel distortion and (b) cochannel distortion.

Type A:q1≠q2≠q3: NA(Q, r) =6冋冉2K2−r冊2−⑀4册 (2.33)

Type B: q1=q2≠q3: NB(Q, r)=3冋冉2Q2−r冊 +⑀2册 (2.34)

where⑀ =mod [r/2], and mod (m/n) is the remainder ofm/n.

Following the same reasoning, the number of mixing products located at cochannel positions, round [(Q+1)/2]≤r≤Q, in Figure 2.23(b) was found to be

Type A1:q1≠q2≠q3and 1≤q1, q2, q3<r:

NA1(Q, r) =6冋冉r−22冊2−⑀41册 (2.35)

Type A2: q1≠q2≠q3andr <q1, q2, q3≤Q:

NA2(Q, r)=6冋冉Q−2r −1冊2−⑀42册 (2.36)

Type A3: q1≠q2≠q3and 1≤q1<r,r<q2≤Q,q2≤q3<q1:

NA3(Q, r) =6 (Q−r) (r−1) (2.37)

Then, adding these three partial contributions, gives

NA(Q, r) =6冋冉r−22冊2−⑀21+冉Q−2r−1冊2−⑀42+(Q −r) (r −1)册

(2.38) where⑀1=mod [(2Q−r)/2], and⑀2=mod [(Q −r +1)/2].

Type B1:q1≠q2and 1≤q1, q2<r:NB1(Q, r)=3冋冉r−22冊+⑀21册

(2.39) Type B2:q1≠q2and r<q1,q2<Q:NB2(Q, r)=3冋冉Q−2r −1冊+⑀22册

(2.40) Then,

NB(Q, r) =3冋冉r−22冊 +⑀21+冉Q−2r−1冊 +⑀22册 (2.41)

where⑀1=mod [(2Q−r)/2], and⑀2=mod [(Q −r +1)/2].

Type C:q1≠q2and q1=r, q2≤Q: NC(Q) =6(Q −1) (2.42) and

Type D:q1=q2=r:ND=3 (2.43) Remembering that products of Type A or B are uncorrelated in phase, and so they must add in power, while the ones of Type C or D are correlated in phase, therefore adding linearly, we are now in condition to derive formulas for approxi- mate small-signal levelM-IMR,ACPRL/U,NPR, andCCPRas a function of the number of tones Q, and the two-tone IMR.Those expressions are presented in Tables 2.2 to 2.5, where IMRstands for the signal-to-intermodulation distortion

Table 2.2 Relations Between Small-SignalQ-Tone and Band-Limited White Gaussian Noise M-IMRand Two-ToneIMR

Q-toneM-IMR M−IMR(Q,r)=3 4

Q2

2NA(Q,r)+NB(Q,r)IMR M−IMRnoise(␻T)=1

8

B2w

␻2T

2 −(Bw+␻h)␻T +(Bw+␻h)2 2

IMR NoiseM-IMR(Q→ ∞)

Table 2.3 Relations Between Small-SignalQ-Tone and Band-Limited White Gaussian Noise ACPRL/Uand Two-ToneIMR

Q-toneACPRL/U ACPRL/U(Q)= 3Q3

4Q3−3Q2−4Q−3 mod (Q/2)IMR ACPRL/U−Noise=3

4IMR NoiseACPRL/U(Q→ ∞)

Table 2.4 Relations Between Small-SignalQ-Tone and Band-Limited White Gaussian NoiseNPR and Two-ToneIMR

Q-toneNPR NPR(Q,r)= Q2

4Q2−8r2+8Qr−38Q+24r+14−2(⑀1+⑀2)IMR NoiseNPR(Q→ ∞) NPRnoise(␻T)= Bw2

−8␻T2+8(␻l+␻h)␻T +4Bw2−8␻h␻l

IMR

Table 2.5 Relations Between Small-SignalQ-Tone and Band-Limited White Gaussian NoiseCCPR and Two-ToneIMR

CCPR(Q)= 3Q3

64Q3−102Q2+56Q+6 mod冉Q2冊IMR

Q-toneCCPR

CCPR= 3 64IMR NoiseCCPR(Q→ ∞)

ratio that would be measured in the same device, when subject to a two-tone excitation having the same average input power as the considered uncorrelated Q-tones. SinceIMRandIP3were already related by (2.15) and (2.16), for a given output power, expressing multitone results in terms ofIP3is now straightforward.

Since the relations presented in Table 2.2 to Table 2.5 were exclusively derived under small-signal regime (imposed by the definition ofIP3), analytical simplicity justified neglecting third order perturbation components in numerators, in compari- son to much stronger linear ones.

For completeness, Tables 2.2 to 2.5 also include results derived from (2.29) when the excitation is a band-limited white Gaussian noise, spanning from ␻l =

␻0 − Bw/2 to ␻h =␻0+ Bw/2, and keeping the same input power level. These can also be interpreted as the limit results that would be obtained for the multitone case if the number of input spectral lines were increased indefinitely, but total average power and bandwidth were kept constant.

Figure 2.24 summarizes these results by showing plots of the various two-tone IMR to M-IMR[Q, (Q + 1)], ACPRL/U(Q), NPR{Q, round [(Q + 1)/2]} and CCPR(Q) ratios, as a function of the number of input tonesQ.

As shown in that figure, the limit of an infinite number of tones (or white Gaussian noise) is almost reached for Q greater than about 10. This serves as an indication of the statistical properties of an uncorrelated multitone signal.

Furthermore, this figure also shows that the referred limits are 6 dB for the IMR/M-IMR(␻h) ratio, 1.25 dB forIMR/ACPRL/U, 7.78 dB forIMR/NPR(␻0), and, 13.29 dB for theIMR/CCPRratio.