Clock jitter, sampling rate, and quantization noise are the most critical factors affecting the SNR and SFDR in ADCs

By Arash Loloee, Senior IC Design Engineer and
Member Group Technical Staff, Texas Instruments

It is not surprising that the world around us is getting more
and more digitized. One reason might be that manipulation, storage, and usage
of data in digital form is more convenient and easier than analog form. That
puts more importance on circuits that convert the analog signals into digital
forms for further processing and use.

When using an analog-to-digital converter (ADC) in any circuit,
the user’s first inquiry is the number of bits (NOB), followed by the sampling
rate. But it is the effective number of bits (ENOB) that will tell you a great
deal about the converter itself. Along with ENOB, the sampling frequency, spurious-free
dynamic range (SFDR), integral nonlinearity (INL), and differential
nonlinearity (DNL) are among the parameters that can guide you when selecting a
prospective ADC.

In this article, I’ll discuss the effect of various parameters
on dynamic behavior, including those that have the most prominent effect on signal-to-noise
ratio (SNR), along with some static behavior of ADCs. I will start by examining
the effects of several key parameters on the SNR, such as jitter, quantization
noise, input signal, and DNL.

Jitter

The uncertainty in the sampling clock results in an error in the
sampled voltage, as shown in Fig. 1, which
depicts the error in sampling voltage caused by clock jitter.

 

Fig. 1:
Sampling clock jitter.

Consider
an input signal of the form of Vin=Vp sin
(2pfin).
The derivative of the input
signal with respect to time is the rate of its change. The maximum change occurs when the cos(2pfin)
is equal to 1, resulting in Equation 1:

 

 TI Equation 1

To ensure that the voltage error from jitter (tj)
is tolerable for a given N-bit ADC with a full-scale voltage (VFS), it has to
be smaller than VLSB/2 (Equation 2), wherein VLSB is defined by VFS/2N:

 

TI Equation 2

For the given sine wave in Equation 1, you can solve for
jitter (tj) using Equation 3:

 

TI Equation 3

I used Vp-p = VFS and Vp=2N–1 VLSB to rewrite the equation.

For a 12-bit ADC with an input frequency of 80 MHz, the
jitter must be smaller than 485 ps not to cause a sampling error.

Now let’s look into the theoretical effect of jitter on the SNR
of the ADC. If Ve-RMS is the worst-case root-mean-square (RMS)
voltage error for tj for the input sine wave, Equation 4 is:

 

TI Equation 4

Equation 5 expresses the contribution of jitter error to SNR
as:

 

TI Equation 5

Quantization noise

Because ADCs represent a distinct range of analog inputs for
a given code, there is bound to be an error associated with each digital code.
Simply put, although a range of analog inputs is assigned to a given code, only
one analog input is represented accurately for the given code. As a result,
there is an error associated for the assigned code to a given analog input.
This error can range from –LSB/2 to +LSB/2.

As an example, in Fig. 2,
all analog inputs in the range of 2.5 to 3.5 are represented with the same digital
code: 0 … 011.

TI SNR/SFDR data converters figure 2

Fig. 2:
Ideal transfer function of an ADC.

Each input in this range has an error that is bigger at the
boundary of the range (with different polarity); the error gets smaller as you
get closer to the center of the range. Plotting the quantization error versus
the analog input range results in the well-known sawtooth plot, as shown in Fig. 3.

 

TI SNR/SFDR data converters figure 3

Fig. 3: Inherent quantization error.

Equation 6 calculates the average noise power (mean square)
of the error over a step, wherein q
is equal to 1 LSB:

 

TI Equation 6

Therefore, Equation 7 gives the SNR
power ratio as:

 

TI Equation 7

If you include quantization noise and overall noise — the
composite RMS DNL — along with input noise to the jitter effect described in Equation
5, you would get Equation 8:

 

TI Equation 8

The first term inside the bracket is the effect of jitter
discussed earlier. The second term is the effect of quantization noise and linearity
error, and the third term is the effect of input noise.

In an ideal case in which jitter, quantization noise, and
input noise are zero, you’ll get the following well-known equation for SNR
(Equation 9):

TI Equation 9

In practice, usually, the first two terms of Equation 8 are
used and the third term is ignored. You can use Equation 8 to calculate jitter
and quantization error from measured SNR under specific conditions. To
calculate the quantization error, a low fin is applied whereby
the effect of jitter is negligible and can be neglected. You can use the measurement
of the resulting low-frequency SNR to calculate ε in Equation 10:

 

TI Equation 10

To calculate jitter, a high-frequency input is applied and
SNR is measured again. In this case, jitter is the main contributor to the degradation
of SNR. Using the new SNR at the high frequency of f, you can readily
calculate tj with Equation
11:

 

TI Equation 11

Input signal

If the input is not full-scale, it will degrade the SNR
accordingly, an effect that is easily quantifiable. For example, if the input
sine wave has an amplitude of Vin, which is a fraction of full
scale, then the ideal SNR will be reduced by 20 log (2Vin/VFS).
If a sine wave with a 2-V amplitude is applied to a 12-bit ADC with a VFS of 5 V,
the input will reduce the theoretical SNR of 74 dB by 1.938 dB and result in an
SNR of 72.062 dB.

Another
dynamic performance parameter for an ADC is the signal-to-noise and distortion
ratio (SINAD or SNRD). SINAD is related to ENOB in the same way that SNR and NOB
are related. By definition, you can calculate SINAD by including the noise and total
harmonic distortion (THD) effect (Equation 12):

 

TI Equation 12

Alternatively,
you can express SINAD in terms of ENOB simply by using Equation 10 and
replacing SNR for SINAD and N for ENOB.

Oversampling

So far, I have
assumed sampling at the Nyquist rate, meaning that the sampling rate is twice
the maximum frequency of the signal being sampled. Now, let’s study the effect
of oversampling — sampling beyond the Nyquist rate — on SNR. For this purpose,
the ratio of sampling frequency (fos) to the Nyquist frequency, 2 fo, is the oversampling
rate (OSR = fos/2f0).

But
first, let’s look at the quantization error, ε, used
in previous equations to calculate SNR. ε has
an equal probability of lying between ±(Δ/2), wherein D is the LSB or simply VFS/2N.
Equation 13 expresses the quantization noise power as:

 

TI Equation 13

Equation 14
gives the noise power that falls into the signal bandwidth (0 to fo) as:

 

TI Equation 14

Equation 14 illustrates
an interesting point: Oversampling decreases quantization noise in the band of
interest, which results in improved SNR. In fact, you can quantify the
improvement in SNR using Equation 15:

 

TI Equation 15

From Equation
15, you can see that SNR improves by 3 dB per octave. So, if OSR = 2, then the SNR
improves by 3 dB; if OSR = 4, the SNR improves by 6 dB.

In Fig. 4, the sampling frequency has
increased by 3 dB per octave; thus, the noise floor has dropped by 9 dB, which
is equivalent to improved SNR by the same amount.

 

TI SNR/SFDR data converters figure 4

Fig. 4:
Reduction of noise floor due to oversampling.

DNL

DNL errors
reduce overall SNR. Below a certain frequency, THD is only dependent on the overall
INL of the ADC; beyond that frequency, the dynamic performance of the converter
comes into the picture.

Let’s
formulate the limit of 1 LSB DNL, which is the condition for not having a missing
code; this is equivalent to a 1-bit reduction in resolution and, thus, a reduction
of SNR by 6 dB. For an n-bit converter with a 1 LSB linearity error, Equation
16 expresses the SNR boundary due to linearity error as:

 

TI Equation 16

This is a first-order
approximation. Although valid only when there is no missing code, Equation 17
gives the SNR degradation caused by DNL as:

 

TI Equation 17

Summary

Clock jitter, sampling rate, and quantization noise are the
most critical factors affecting the SNR and SFDR in ADCs. Factors like jitter
and sampling rate can affect the expected and theoretical values of many
important parameters such as SNR. By understanding the factors affecting SNR
and utilizing some method to quantify the effect, one can better estimate the
SNR for a given ADC. Additionally, we can try to optimize the design to obtain
the target specifications after understanding and quantifying those factors’
effects on performance. 

 

About the author

Arash Loloee is a senior integrated-circuit design
engineer and Member Group Technical Staff at Texas Instruments. His
responsibilities include designing and modeling transistor- to system-level projects.
He was a lecturer from 2005 to 2010 at the University of Texas at Dallas, where
he taught various analog and digital courses. He has a bachelor’s degree in
physics from the University of North Texas and master’s and Ph.D. degrees in
electrical engineering from Southern Methodist University. For questions about
this article, please
contact Arash at ti_arashloloee@list.ti.com.

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