The Noise Power Ratio - Theory and ADC Testing

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1 The Noise Power Ratio - Theory and ADC Testing FH Irons, KJ Riley, and DM Hummels Abstract This aer develos theory behind the noise ower ratio (NPR) testing of ADCs. A mid-riser formulation is used for mathematical simlicity. Simulated results, using DAC generated signals, suggests that the uniformly distributed signal is easier to imlement and is more sensitive to amlitude deendent distortions.. Introduction Historically, analog-to-digital converters (ADCs) have been characterized for static and dynamic error erformance using ram and sinewave test signals resectively. However, many modern alications that deal with wideband signals such as telehone suer grous, multilexed data channels, cellular base station receivers, and others, require some form of wideband, random (or seudo-random) noise structure to be used as characterization signals to check ADC resonse to such signals. Since wideband signals induce intermodulation distortion rather than harmonic distortion as observed with single tones, it is often felt that wideband testing rovides more reassuring results for these alications. The noise ower ratio (NPR) test has been known for several years [, ]. Test equiment was develoed to generate noise signals over standard CCIR bands along with secific notch filters to shae the random signal rior to samling. The construction of narrow-band dee notches ( 8) db is rogressively more difficult to achieve at higher frequencies as both ADC samling frequency and resolution has imroved in recent years. Tyical samling frequencies have gone from MHz to several hundred MHz in the last 5 years. Imlementation difficulties have always rovided a negative imact uon the desire and ability to erform NPR testing. Current technology is showing that digital-to-analog converter (DAC) generated test signals are becoming a reality. In addition, the need for broadband testing rocedures has been recognized in the current draft of ADC Standard 4, [3]. Wideband noise ower ratio testing is included through the use of rogrammed DACs to erform both waveform generation and notch filtering functions. This aer resents a review of the theory and tyical results for two random broadband signals, namely: notched Gaussian or Uniform noise distributions.. Noise Power Ratios Equations for the NPR measure can become quite messy according to models used for the analysis. A mid-riser transfer function alignment is used to obtain symmetry around zero for a biolar signal having the range L V. The non-symmetrical (for an even number of states) midtread alignment comlicates analysis and the difference is not worth the effort required to erform the analysis. Consequently, the mid-tread model is not discussed in this aer... The Uniformly Distributed Signal The uniform robability distribution function rovides a simle basis to develo rocedures used to determine equations for NPR measures on ideal limiting quantizers. Let the inut to the quantizer, x, be given by a random signal uniformly distributed over the range A V where A may be greater or less than the quantizer limit, L. The quantizer outut consists of discrete states (binary codes) in least significant bits (LSBs) such that the transfer function is a stair ste ram about a straight line through the midoint of each riser. The difference between the discrete outut and the hyothetical straight line is considered to be quantizer error. For convenience, the error, q(x), is referred to the inut by multilying by a Volts/LSB constant to facilitate estimation of inut and distortion mean squared values. A curve is shown in Fig. for N = 4 bits, L = V, and Q = L= N (V/LSB). The curve shows that the V full scale inut range sans the interval where error lies between Q=. It is also clear that there are exactly N sub-regions in this range with half of the states ositioned on each side of x =. The error, q(x), is a iece-wise straight line connected function where each line has a sloe of?. Now let U (x) reresent the robability density function for the uniformly distributed signal, x, over the range A

2 Error q/q (LSB) Inut x (V) Figure. An error curve for an ideal quantizer. as given in () where u(x) is the unit ste function. U (x) = [u(x + A)? u(x? A)] () A The exected mean square inut is then obtained from (). E(x ) = dx x?a A = A 3 = (V in) rms () The exected mean square error deends uon the value of A chosen for the inut distribution. For A < L, the ideal error is in the range Q= and a single integral determines the error estimate. In the following derivation, a continuous range of A values is chosen using an integer variable, m = Integer Part(A=Q), and a continuous variable, = Fractional Part(A=Q). These variables have allowable ranges, m N? and <.. A = (m + )Q < L: E(q ) = A E = dx q?a A A m?z (k+)q k= kq Z (m+)q dx q dx(q=? x + kq) + dx(q=? x + mq) mq E(q ) = Q m (m + ) + 3 8? (:5? )3 or E(q ) = Q [ + F (m; )] where (3) 4(? :5)(? ) F = (m + ) The result in (3) shows that exected squared error only equals the classic Q = result for A values that coincide with each riser transition and mid-tread value. The deviation from Q = is not significant for large A, but is significant for small A, esecially for m = or m =, and for the number of bits, N, less than 5 or 6. In the following derivation for A > L, the integral includes the full range of the quantizer so (3) is used with = for the ortion of the integral covering x L.. A > L E(q ) = dx q reduces to?a A Z?L A E = dx(+q=? x? L)?A + AQ + dx(?q=? x + L) L Z E(q ) = Q + A dx(q= + x? L) A L = Q [ + G(A; L)] where (4) 3(A? L) (A? L) 4(A? L) G = + + A Q 3Q This last result shows that for signals exceeding the limiter range, the exected mean square error consists of two arts due to the in- and out-of-range errors. Equations (3) and (4) are combined to obtain the uniform NPR measure for all A > as given in (5) where A = 3(V in ) rms and Q = L= N. = NP R U = E(x ) E(q ) (A=Q) + F (m; )u(l? A) + G(A; L)u(A? L) The inut at maximum NPR in (5) is seen to occur when A = L? with the following result. max NP R U = L =3 L Q = = N or Q (5) max NP R U = log ( N ) = 6:N db (6) Assuming such a test can be created, it has the merit that it dislays the aarent broadband resolution of the quantizer in a measure roortional to the number of bits. 3. Simlified Form for A > L A simlified result can be develoed for (5) by setting the quantizer range, L, to coincide with the last zero error values rather than at the Q= boundaries shown in Fig.. This requires using an odd number of states to set the quantization interval to Q o = L=( N? ). The exected squared error for A < L is still the same as in (3), but at A = L, = :5 instead of as obtained for the quantization interval, Q e = L= N. Equation (7) is now obtained

3 NPR (db) with a single function over the full range of inut, the integration to determine exected values is the same for either large or small signals. Following the above rocedure, the analysis for large signal behavior ( x > L) is simler when the error characteristic, q(x), is defined with the quantizer range, L, set to coincide with the last zeros of q(x). Let G (x) reresent the robability density function for a Gaussian signal, x, over the range () as given in (9). (8) (5) G = e?:5(x=) (9) Figure. NPR for uniform noise signal. for A > L. Z E(q ) = Q o + A dx(x? L) A L = Q o 4(A? L)3 + Q o A This result is considerably simler than that given in (4). The useful result of the NPR test is to determine the eak NPR for a secific ADC and so normally the inut limit, A, is much greater than Q and the NPR is greater than db. As A becomes greater than Q, the factor, F (m; ) in (5), tends toward zero and can be ignored when comared to one. A simlified form of the NPR measure is now obtained as given in (8) for A > Q o. NP R U = E(x ) E(q ) (A=Q o ) (7) + 4(A?L)3 Q o A u(a? L) (8) The lot shown in Fig. comares (8) to (5), for an ideal 8- bit ADC, to show the valid range of the aroximate result. The dot on the db line indicates the inut coinciding with A = Q o. The lot shows that there is no signifigant difference between the two results for rms inut voltages that are greater than a few quantization intervals. For normal resolution (> 6 bits), (8) is quite adequate and rovides a simler model for the uniform signal NPR measure and it is the equation used in Standard 4... The Gaussian Distributed Signal Results from the analysis for the uniformly distributed signal rovide background for roceeding with an analysis for the NPR measure of a Gaussian distributed random signal. Since the Gaussian distributed signal is reresented The exected mean square inut is then given in (). E(x ) =? dx x G (x) = (V in ) rms () To obtain the exected mean square error it is necessary to break the integral into two arts to handle the iecewise connected descrition of q(x). Again, q(x) is taken as an odd-symmetrical function of x with L set at the last zeros in the quantizing range. Thus the quantization interval is based uon N? states and Q = Q o = L=( N?). The exected error is found after use of the following change of variable and integration-by- arts. v = x= y(v) = e?:5v = dy =?vy(v) () dv E(q ) =? dx G(x)q (x) Z L?Q= = dx G q + dx G (x? L) L?Q= = N + N () In (), N is the squared error for signals within the quantizer region (jqj Q=) and N is the squared error for signals outside the quantizer range (j q j Q=). For greater than a few quantization intervals, or for larger resolution (bits), N is aroximately Q = as found for the uniform signal analysis. The exact functions are given in (3) and (4). Note in (), the integrals are searated at the oint x = L? Q= to simlify analysis. Z N? L?Q=? N = dx G k= = [u(x? kq)? u(x? (k + )Q)] N??Z kq+q k= kq (Q=? x + kq) dx e?:5(x=) (x? kq? Q=)

4 NPR (db) (7) (5) Figure 3. NPR for Gaussian noise signal. and with v = x= we get: N = N??Z (k+)q= k= kq= dvy(v) [(v)? Q(k + :5)v + (k + :5) Q ] N?? = Q fy((k + )Q=)k=Q + (3) k=?y(kq=)(k + )=Q + Z (k+)q= [(=Q) + (k + :5) ] dvy(v)g kq= The second integral, N, contributes the dominant distortion to the squared error measure for > L=. N = L?Q= dx e?:5(x=) and with v = x= we get: N = (x? L) dvy(v)[(v)? Lv + L ] (L?Q=)= = f(l + ) dvy(v) + (L?Q=)=?(L + Q=)y((L? Q=)=)g (4) Equations (3) and (4) are then used to form the desired NPR measure. NP R G = E(x ) E(q ) = N + N (5) The result in (5) is exact and is valid for any number of bits or inut signal level. The result given in Standard 4 is simlified for the case where the range, L, is assumed to be much larger than the quantization interval, Q = L=( N? ), and the integral, N, aroximates the ideal Q = value. In addition, N is aroximated by the limit given in (6) with subsequent reresentation of the NP R as in (7). L >> Q=: and N a = ( + L ) L= NP R G dvy(v)? Ly(L=) (6) Q = + N a (7) Figure 3 shows a comarison of (7) to (5) for an ideal 8-bit ADC. The only difference between the two curves is seen for small inut. For any given ADC, the NPR test is usually only concerned about the eak value and the ower level at which it occurs. The maximum NPR and the corresonding inut ower, for the Gaussian distribution, is difficult to obtain from (5) in closed form as was done for the uniformly distributed signal. Standard 4 lists a set of values obtained by numerical methods. The results are listed for a normalized rms inut level, = =L, and for bit values ranging from 6 through. Tabulated results show that eak values increase by slightly less than 6 db/bit of resolution for the Gaussian distributed signal. 3. Results and Conclusions The above theoretical relationshis are tested via simulation through the use of a DAC to generate notched wideband random signals. Examles shown use an 8-bit ADC with a V quantizing range and a comression characteristic that yields odd harmonic distortion with a resultant 4:3 db SFDR. The DAC has -bits with a linear range of V for generating the uniformly distributed signal and a 3 V range for the Gaussian distributed signal. Random signals are generated from -bit quantization of notched sectra. An examle is shown on the inut curve of the FFT magnitude sectrum given in Fig. 4. Three notches aear, one bandsto notch near F S =4, a lowass notch around F S =, and the conjugate notch at 3F S =4. The lowass notch rovides Nyquist filtering to revent aliasing and the bandsto notch is used to measure distortion for the NPR test. Exeriments have shown that a single lowass notch is quite often sufficient. 48 samles were used for each waveform for these examles. The NPR ratio is formed by summing ower in the notched bins to form the distortion measure and then summing ower in an equal number of of unnotched bins to form the signal measure. The NPR is then the ratio of signal-to-distortion measures. Figure 5 shows results for a uniformly distributed NPR test with the ADC resonse comared to theory for an 8-bit converter. The NPR is

5 6 45 Inut (dbfs) Outut (dbfs) Vrms=.5L (V) FFT Bin NPR (db) Figure 4. Tyical ADC inut and outut sectra. Figure 6. 8-bit examle Gaussian resonse. NPR (db) Figure 5. 8-bit examle Uniform resonse. lotted against the rms signal actually alied to the ADC. This inut rms voltage is calculated from the average of the squared samles of the inut to the DAC. Note the eak of the NPR measure is about 5:6 db below the eak of the theoretical curve. This corresonds to the nearly 6 db loss in SFDR for the 8-bit ADC model that was used for the simulation. The failure of the NPR to fit the downward fall of the theoretical resonse is due to two factors. First is the notching of the sectra which causes overshoot. What starts out as a uniform distribution ends u exceeding eakto-eak limits set on the function by as much as to 5 % deending uon notch widths and filter shaes. Second is that when signals begin to reach, or exceed, the DAC range, then further distortion occurs and the NPR dros faster. It has been observed that when the NPR measure does not fit the theoretical curve for small signals, internally generated noise, such as samle-time jitter, clock noise, and other broadband effects, reduce effective bits and translate the NPR curve downward. Fig. 6 shows results of the NPR test on the same ADC model for a Gaussian distributed signal. Here the DAC range is 3 V. The result does not show any loss at the eak at all! Caution must be exercised when interreting ADC erformance from NPR measures. The Gaussian signal sends 65 % of the time between a sigma range, whereas the uniform signal stresses all ranges uniformly and sends as much time at large as well as small signal levels. Thus the uniform signal shows losses due to comression effects whereas the Gaussian signal does not. Finally, it should be noted that use of the aroximate NPR equations (8) and (7) are ractical for the NPR grahs because they can be easily comuted. In conclusion, this aer has resented suorting theoretical develoments for the NPR test of ADCs for both Gaussian and Uniform signals. The uniform distribution aears to offer easier imlementation for the DAC signal range versus the ADC under test, and it also rovides more sensitive resonse to amlitude deendent harmonic distortion effects. References [] Kester, WA, Test Video A/D Converters Using Dynamic Conditions, EDN, Aug 98 [] Gray, GA, & Zeoli, GW, Quantization and Saturation Noise Due to Analog-to-Digital Convertersion, IEEE Trans on Aerosace & Elec Systems, Jan 97, - 3 [3] Standard 4, Sub-committee on ADC testing, IEEE Technical Committee, TC-, Draft in rocess

DSP IC, Solutions. The pseudo-power entering into the adaptor is: 2 b 2 2 ) (a 2. Simple, but long and tedious simplification, yields p = 0.

DSP IC, Solutions. The pseudo-power entering into the adaptor is: 2 b 2 2 ) (a 2. Simple, but long and tedious simplification, yields p = 0. 5 FINITE WORD LENGTH EFFECTS 5.4 For a two-ort adator we have: b a + α(a a ) b a + α(a a ) α R R R + R The seudo-ower entering into the adator is: R (a b ) + R (a b ) Simle, but long and tedious simlification,