# Errata

This is an errata list for the course book. Please report suspected errors to the course representatives to make the list more complete.

[Data Converters], p.19, equation 1.16*Problem*: Wrong symbol*Solution*: Psine should be Ptrian

[Data Converters], p.20, equation 1.21*Problem*: Equation 1.21 is completely wrong, and probably not the one used to generate figure 1.16.*Solution*: The correct expression for the ENB can be found in the lecture slides.

[Data Converters], p.23, equation 1.25

*Problem*: The integrand of the inner integral in Equation 1.25 is wrong.* Solution*: It should be the voltage squared given by Equation 1.24.

[Data Converters], p.24, Fig. 1.19*Problem*: y-axis is wrong*Solution*: It should be sqrt(kT/C), and not kT/C

[Data Converters], p.28, in the text

*Problem*: ...coherent sampling for which an integer number of clock cycles, k, fits into the sampling window...* Solution*: ...coherent sampling for which an integer number of signal periods, k, fits into the sampling window...

[Data Converters], p.35, equation1.40

*Problem*: wrong expression of H_{S&H}* Solution*: romeve the leading "j" in the expression

[Data Converters], p.42, equation 1.55 and 1.56

*Problem*: The equations and texts do not correspond* Solution*: Equation 1.55 is the backward transformation, while 1.56 is the forward transformation and not the other way around.

[Data Converters], p.57, the second paragraph of the text

*Problem*: Reference to equation 2.2 is wrong.* Solution*: The correct reference should be to equation 2.1

[Data Converters], p.57, figure 2.8

*Problem*: Even though the text in the last paragraph about the DNL/INL talks about the running sum of the correlated part of the DNL, figure 2.8 is referring to the correlated INL.* Solution*: Swapping INL for DNL in figure 2.8 makes more sense.

[Data Converters], p.57, equation 2.3

*Problem*: This equation for transition points after correction (for gain and offset) X'(k) does not make sense. Setting kos=0 and all DNLs=0 gives that all transition points should be X'(k)=0 for all k.* Solution*: Better equations for DNL and INL are presented in the lecture slides.

[Data Converters], p.58, example 2.1

*Problem*: Implementing equation 2.5 with the described values does not generate the plots in figure 2.9 and 2.10.* Solution*: The MATLAB code has a factor 800 multiplied to the equation for y. This factor is missing in the example text.

[Data Converters], p.59, example 2.1, in the figures

*Problem*: Figure 2.9 and 2.10 are probably not the correct figures for this example

*Solution*: The output spectrum should contain the 2nd, 3rd and 4th order harmonic tones of the input tone since both a=-0.01, b=0.01 and c=0.02 are set in the polynomial describing the nonlinearity. The reason for this problem is that in the MATLAB code an 11 bit quantization is implemented rather than a 12 bit quantization like it says in the example. A division by 2^(N-1) is done rather than by 2^N in the code, so the harmonics drown in quantization noise.

[Data Converters], p.68-69

*Problem*: Two-tone intermodulation distortion is given the acronym IMD2; however, IM2 (IM3) is commonly used to indicate the second-order (third-order) intermodulation distortion, i.e., the distortion caused by a quadratic (cubic) non linearity in the transfer function. * Solution*: Be aware of the above.

[Data Converters], p.71

*Problem*: In the definition of ERBW, it is not clear whether the amplitude of the input signal is arbitrary or not.

[Data Converters], p.119, equation 3.29.* Problem*: Equation error

*Solution*: The equation shall be multiplied by 1/2 to be correct.

[Data Converters], p.120, equation 3.34.* Problem*: Equation error

*Solution*: Caused by using the erroneous result from 3.29.

[Data Converters], p.121, equation 3.41.*Problem*: Equation error

*Solution*: It should be DNLmax = sqrt(2^n)*Delta I_r/Delta I_u.... Furthermore, three lines below, the DNL error with unary selection should be Delta I_r/Delta I_u, without a factor 2.

[Data Converters], p.143, section 4.2, 2nd paragraph

*Problem*: 1/2 LSB = Vfs/(2^n+1)

*Solution*: Should be 1/2 LSB = Vfs/(2^(n+1))

[Data Converters], p.144, eqs. 4.8-4.9

*Problem*: Equation errors

*Solution*: In 4.8, "erf" should be replaced with "erfc". Furthermore, 4.9 is an expression of the time constant T_C, and not of the delay T_D (see Kaupp, JSSC Aug. 1989, cited in p. 207).

[Data Converters], p.145-146, example 4.2, calculation error

*Problem*: The final calculation of the jitters is wrong

*Solution*: The noise should be divided by the slope to get 41 fs and 14 fs.

[Data Converters], p.148, equation 4.13* Problem*: Equation error.

*: In the equation, it looks like Vref- is multiplied by the rest of the equation. Actually, there should be an addition (+ sign) between Vref- and the rest.*

Solution

Solution

[Data Converters], p.156, equation 4.20* Problem*: The equation results in to low current (80 uA).

*: The expression for the bias current should be multiplied by the dynamic gain A0. Thus the correct answer should be 1.6 mA. For the 7-bit converter the current would be 204.8 mA.*

Solution

Solution

[Data Converters], p.160, second paragraph of the text.

*Problem*: Reference to fig 4.12

*Solution*: Should be a reference to fig. 4.11.

[Data Converters], p.169, Fig. 42(b).

*Problem*: Curve Vo(i+1)

*Solution*: Should have zero-crossing at Vref(i+1)

[Data Converters], p.172, equation 4.28 and figure 4.24

*Problem*: Errors in the equation

*Solution*: The eout,i in the figure is called eI,i in the equation. Furthermore, the multiplication Rout*RT,i should be a division Rout/RT,i.

[Data Converters], p.176, 6 lines after equation 4.32

*Problem*: Images are said to be located at k*f_s/(2N) +- f_in

*Solution*: Images are located at k*f_s/N +- f_in

[Data Converters], p.180, figure 4.30* Problem*: The sampled and held voltage does not correspond to the output code, even in the example with the correct ouput code.

*: The correct S&H voltage should be 0.4296875.*

Solution

Solution

[Data Converters], p.186, figure 4.35* Problem and solution*: The bits in the second stage og the pipeline converter at time n+1 should be b7-b6, not b9-b8.

[Data Converters], p.217, in the text

*Problem*: Notations in the figure and text mismatch.* Solution*: IB1 in the text refers to I1 in the figure and IB2 in the text refers to I2 in the figure.

[Data Converters], p.218, equation 5.6.* Problem*: An error in the equation.

*Solution*: The dVin should be divided by dt, not by a comma!

[Data Converters], p.219, figure 5.9.* Problem*: The figure is empty.

*Solution*: Run the MATLAB file Ex5_1.m to get the figure.

[Data Converters], p.220, equation 5.10

*Problem*: The noise coefficient gamma should not be 2/3 for a bipolar transistor.

*Solution*: The gamma for bipolar transistors are typically 1/2, 2/3 is valid in the ideal case for a MOS transistor.

[Data Converters], p.220, in the fourth paragraph of the text

*Problem*: It is claimed that the small signal output impedance of the current source will generate noise, which is a fundamental mistake.* Solution*: It is a well known fact that small signal parameters do not generate any noise.

[Data Converters], p.221, equation 5.12

*Problem*: The noise coefficient gamma is used as a noise excess factor.* Solution*: The gamma of equation 5.12 is not the same as the gamma of equation 5.11, in the lecture slides we have denoted this excess noise factor gamma'.

[Data Converters], p.231, in the last paragraph of the text

*Problem*: The conclusion for the noise from generator (3) in Fig. 5.18 is wrong. *Solution*: The noise sampling on the two capacitors does not cancel the noise from generator (3). Rather on the contrary, the noise is actually amplified by 2.

[Data Converters], p.235, figure 5.20

*Problem*: The arrows in the picture point to nothing.* Solution*: The pictures are corrected in the lecture slides.

[Data Converters], p.236, equation 5.26

*Problem*: Error in the equation.* Solution*: Instead of Vin-Vthp it should be Vin-|Vthp|.

[Data Converters], p.239, in the first paragraph of the text.

*Problem*: The reason for the bootstrapping is not properly explained.* Solution*: Since the gate-source voltage of the switch is almost independent of the input voltage Vin, the linearity of the switch is greatly improved.

[Data Converters], p.239, figure 5.24.

*Problem*: The schematic has an error* Solution*: The gate of M1 should be connected to the gate of Md2 (source of Md1) for this circuit to work properly.

[Data Converters], p.240, in the text, paragraph 3.

*Problem*: The reason for not using a PMOS transistor is not clearly explained.* Solution*: The reason is that with a PMOS transistor, Vg of M1 would need to become larger than Vdd during the switch on-phase. To avoid that, a more complex biasing scheme would be needed.

[Data Converters], p.243, section 5.8, third sentence

*Problem*: The operation regions of the transistor has been mixed up.* Solution*: "As a matter of fact, a MOS transistor in the

**saturation**region....

[Data Converters], p.246, figure 5.28

*Problem*: The figure is not referred to anywhere in the text.* Solution*: The figure belongs to example 5.4. It shows the transconductance of the V-I converter, with the input voltage on the y-axis and the output current on the x-axis.

[Data Converters], p.255, figure 6.2.

*Problem*: The (a) figure is unclear.* Solution*: What looks like a buffer in the figure is meant to be an integrator. There should be some integrator sign on the buffer in the feedback path.

[Data Converters], p.256, figure 6.3.

*Problem*: The (a) and (b) figures are unclear.* Solution*: What looks like buffers in the figures are meant to be integrators. There should be some integrator sign on the buffers in the forward path.

[Data Converters], p.257, paragraph 3, sentence 3.

*Problem*: The blocks have been mixed up in the text.* Solution*: "A second processing block B(z)...

[Data Converters], p.260, paragraph 3.

*Problem*: An error in the text* Solution*: "..., the number of bits sent to the digital filter is the

**upwards**rounding of

**log2(k+1)**;therefore, ..."

[Data Converters], p.260-262, example 6.1.

*Problem*: With Vfs=1V, the input signal with amplitude 0.85 V would cause overload of the quantizer of the Delta-Sigma ADC. * Solution*: In the simulation leading to the results in figure 6.8, Vfs=2V was used, so clipping was avoided. However, that makes the calculation of the quantization noise power and PSD of the quantization noise close to f=fs/2 wrong. They should be recalculated with delta=1/4.

[Data Converters], p.265, paragraph 2, sentence 2.

*Problem*: Strange choice of words using dump instead of damp.* Solution*: "..., it is necessary to

**damp**one of the two integrators..."

[Data Converters], p.268, equation 6.26.

*Problem*: C_{1} [... V_{out}(nT)/A_{0}].* Solution*: C

_{1}[... V

_{out}(nT+T)/A

_{0}].

[Data Converters], p.272-273, in the text.

*Problem*: Different notations have been used for the same thing.* Solution*: Vstep and DVout is probably the same thing, and t and td is also the same thing.

[Data Converters], p.274, in the text and figure 6.17.

*Problem*: The SR limitation is not the same in the text as in the figure.

[Data Converters], p.276, equation 6.41.

*Problem*: The equation does not tale into account that the quantization noise is shaped.* Solution*: The r.h.s of the inequality is given by the shaled noise power in eq. 6.23 (in the case of a 2nd order modulator).

[Data Converters], p.360, figure 8.1.

*Problem*: The reference voltage connected to C1 and C2 is called VB in the text, but Vref in the figure. There is also a reference to Vdac in the text, not shown in any figure.* Solution*: The notation in the figure is wrong since Vref is used to denote the DAC voltage. Probably Vdac is the same as Vref.

[Data Converters], p.363, equation 8.5 and 8.6.

*Problem*: There are two different notations for Vref, Vdac and Vref.* Solution*: It should be Vref in equation 8.6 instead of Vdac.

[Data Converters], p.370, equation 8.9.

*Problem*: Error in the equation.* Solution*: Where it says dG,i it should probably be (1+dG,i).

[Data Converters], p.374, equation 8.13.

*Problem*: It is a bit unclear how to arrive at equation 8.13. At least, it seems like the fact stated in the text, X1+X2=1 is not used.

[Data Converters], p.379, equation 8.20.

*Problem*: Error in the equation.* Solution*: Firstly, the power of the full scale sine wave, mentioned in the text should be (M*X)^2/8. That makes equation 8.20 wrong. Secondly, the OSR should be in the nominator of the SNR expression, so that the SNR increases when OSR increases.

[Data Converters], p.383, fig. 8.22.

*Problem*: The plots for elements #4 and #6 in (a) (rotation approach) are wrong.

[Data Converters], p.387, in the text.

*Problem*: "...the total error given by of the elements..."

*Solution:* "...the total error given by the elements..." or "...the total error of the elements..."