1. Summary

2. Ideal DAC
Essentially a digitally controlled voltage, current or charge source
Example below is for unipolar DAC
Ideal DAC does not introduce quantization error!

3. The Reconstruction Problem
As long as we sample fast enough, x(n) contains all information about x(t)
fs > 2·fsig,max
How to reconstruct x(t) from x(n)?
Ideal interpolation formula
Very hard to build an analog circuit that does this…

4. Zero-Order Hold Reconstruction
The most practical way of reconstructing the continuous time signal is to simply "hold" the discrete time values
Either for full period of Ts or a fraction of Ts
What does this do to the signal spectrum?
We'll analyze this in two steps
First look at infinitely narrow reconstruction pulses

5. Useful Lemmas

6. Dirac Pulses xd(t) is zero between pulses
Note that x(n) is undefined at these times
Multiplication in time means convolution in frequency
Resulting spectrum
The details are given in the next slide.

7. Cont’d

8. Spectrum
Spectrum of Dirac signal contains replicas of X(f) at integer multiples of the sampling frequency

9. Finite Hold Pulse
Consider the general case with a rectangular pulse 0 < Tp ≤ Ts
xp(t) is obtained from convolving the Dirac sequence xd(t) with a rectangular unit pulse hp(t).
xp(t)=xd(t)*hp(t) (the proof is given in the next slide)
Spectrum follows from multiplication with Fourier transform of the pulse

10. Cont’d

11. Envelope with Hold Pulse Tp=Ts

12. Envelope with Hold Pulse Tp=0.5·Ts

13. Example

14. Reconstruction Filter
Also called smoothing filter
Same situation as with anti-alias filter
A brick wall filter would be nice
Oversampling helps reduce filter order

15. Summary Must obey sampling theorem fs > 2·fsig,max
Usually dictates anti-aliasing filter
If sampling theorem is met, continuous time signal can be recovered from discrete time sequence without loss of information
A zero order hold in conjunction with a smoothing filter is the most common way to reconstruct
May need to add pre- or post-emphasis to cancel droop due to sinc envelope
Oversampling helps reduce order of anti-aliasing and reconstruction filters

16. Static Specifications
Maloberti, Data Converters, pp

17. Static Specifications
Static deviations of transfer characteristics from ideality
Offset
Gain error
Differential Nonlinearity (DNL)
Integral Nonlinearity (INL)
Monotonicity
Missing code
Useful references
Analog Devices MT-010: The Importance of Data Converter Static Specifications
"Understanding Data Converters," Texas Instruments Application Report LAA013, 1995

18. Offset and Gain Error
Conceptually simple, but lots of (uninteresting) details in how exactly these errors should be defined
General idea (neglecting staircase nature of transfer functions):

19. ADC Offset and Gain Error
Definitions based on bottom and top endpoints of transfer characteristic
Endpoints: ½ LSB before first transition and ½ LSB after last transition
Offset is the deviation of bottom endpoint from its ideal location
Gain error is the deviation of top endpoint from its ideal location with offset removed
Both quantities are measured in LSB or as percentage of full-scale range

20. DAC Offset and Gain Error
Same idea, except that endpoints are directly defined by analog output values at minimum and maximum digital input
Also note that errors are specified along the vertical axis

21. Comments on Offset and Gain Errors
Definitions on the previous slides are the ones typically used in industry
IEEE Standard suggest somewhat more sophisticated definitions.
Technically more suitable metric when the transfer characteristics are significantly non-uniform or nonlinear
Generally, it is important to build a converter with very good gain/offset specifications
However, since gain and offset affect all codes uniformly, these errors tend to be easy to correct
E.g. using a digital pre- or post-processing operation
Also, many applications are insensitive to a certain level of gain and offset errors
E.g. audio signals, communication-type signals, ...
More interesting aspect: linearity
DNL and INL

22. Differential Nonlinearity (DNL)
In an ideal world, all ADC codes would have equal width; all DAC output increments would have same size
DNL(k) is a vector that quantifies for each code k the deviation of this width from the "average" width (step size)
DNL(k) is a measure of uniformity, it does not depend on gain and offset errors
Scaling and shifting a transfer characteristic does not alter its uniformity and hence DNL(k)
Let's look at an example

23. ADC DNL Example

24. ADC DNL Example (cont’d)
What is the average code width?
ADC with perfect uniformity would divide the range between first and last transition into 6 equal pieces
Hence calculate average code width (i.e. LSB size) as
Now calculate DNL(k) for each code k using

25. Result Positive/negative DNL implies wide/narrow code, respectively
DNL = -1 LSB implies missing code
Impossible to have DNL < -1 LSB for an ADC
But possible to have DNL > +1 LSB
Can show that sum over all DNL(k) is equal to zero

26. A Typical ADC DNL Plot
People speak about DNL often only in terms of min/max number across all codes
E.g. DNL = +0.63/-0.91 LSB
Might argue in some cases that any code with DNL < -0.9 LSB is essentially a missing code
Why ?

27. Impact of Noise
In essentially all moderate to high-resolution ADCs, the transition levels carry noise that is somewhat comparable to the size of an LSB
Noise "smears out" DNL, can hide missing codes
Especially for converters whose input referred (thermal) noise is larger than an LSB, DNL is a "fairly useless" metric

28. Missing Code in ADC
Code 101 is missed.

29. DAC DNL Same idea applies Find output increments for each digital code
Find increment that divides range into equal steps
Calculate DNL for each code k using
One difference between ADC and DAC is that DAC DNL can be less than -1 LSB
How ?

30. Non-Monotonic DAC In a DAC, DNL < -1LSB implies non-monotonicity
How about a non-monotonic ADC?

31. Integral Nonlinearity (INL)
General idea
For each "relevant point" of the transfer characteristic, quantify distance from a straight line drawn through the endpoints
An alternative, less common definition uses a least square fit line as a reference
Just as with DNL, the INL of a converter is by definition independent of gain and offset errors

32. ADC INL Example
"Straight line" reference is uniform staircase between first and last transition
INL for each code is
Obviously INL(1) = 0 and INL(7) = 0
INL(0) is undefined

33. ADC INL Example (cont’d)
Can show that
Means that once we computed DNL, we can easily find INL using a cumulative sum operation on the DNL vector
Using DNL values from last lecture, we find

34. Result

35. A Typical ADC DNL/INL Plot
DNL/INL signature often reveals architectural details
E.g. major transitions
We'll see more examples in the context of DACs
Since INL is a cumulative measure, it turns out to be less sensitive than DNL to thermal noise "smearing"

36. DAC INL

37. DAC INL Same idea applies
Find ideal output values that lie on a straight line between endpoints
Calculate INL for each code k using
Interesting property related to DAC INL
If for all codes |INL| < 0.5 LSB, it follows
that all |DNL| < 1 LSB
A sufficient (but not necessary) condition
for monotonicity