Chapter 2 Overview of Analog-to-Digital Converters
2.2 Fundamental Aspects of A/D Converters
2.2.3 Performance Limitations
Before proceeding, it is required to know the performance metrics for determining the transfer response of the data converters. In this section, some commonly used terms characterizing the performance of data converters are introduced as below.
2.2.3.1 Resolution
The resolution of an ADC is defined to be the number of the distinct input segments corresponding to the different output word. It also indicates the minimal difference of the input signal that can be recognized by the ADC. An N-bit resolution ADC means that the converter can resolve 2N distinct input segments. We can find that high resolution ADCs can resolve smaller segments of the input signal than low
resolution ADCs. This quantity does not mean actually the accuracy of the converter, but instead it usually refers to the number of output bits.
2.2.3.2 Signal-to-Noise Ratio (SNR)
The signal-to-noise ratio (SNR) is the ratio of the signal power to the output noise power. The SNR includes the quantization noise and other circuits noise excluding the harmonic components of the input signal. If it is assumed that the input signal is a sinusoidal waveform between 0 and V , then the RMS value of the sinusoidal wave ref
is equal to Vref / 2 2
( )
. If we only consider the quantization noise of the ADCs, theHowever, the SNR decreases from the best possible value for reduced the input signal levels [4].
2.2.3.3 Signal-to-Noise plus Distortion Ratio (SNDR)
The signal-to-noise plus distortion ratio (SNDR) is often used to measure the performance of an ADC. When a sinusoidal signal is applied to an ADC, the output spectrum generally contains a single tone at the fundamental frequency. Due to distortion, the output spectrum also contains several tones at the harmonic frequency, known as harmonic distortion. As a result, the SNDR of the ADC is defined as the ratio of the signal power at the fundamental frequency to the total power of non-ideal effects, including the harmonic distortion, quantization noise and other noise sources
2.2.3.4 Spurious Free Dynamic Range (SFDR)
The spurious free dynamic range is defined as the power ratio of the input signal to the largest distortion component. In a fully differential signal system, generally the largest distortion component is the 3rd harmonic term.
For more clearly figuring out the difference among SNR, SNDR and SFDR, a spectrum diagram is shown in Figure 2.5, where S is the fundamental frequency of the input signal, D are the distortion components and N is the noise floor.
Power Spectrum
f S
N
D
SFDR
fin 2fin 3fin
Figure 2.5 The spectrum diagram with signal, distortion and noise.
The SFDR is depicted in Figure 2.5, and the SNR and SNDR are depicted as below respectively.
S S
SNR SNDR
N N D
= =
+ (2.7)
2.2.3.5 Effective Number of Bits (ENOB)
Another specification often used to describe the ADC’s performance is the effective number of bits (ENOB). Different from resolution, ENOB indicates the ADC’s accuracy in a specific input frequency and sampling rate, and it can be expressed from SNDR as follow:
-1.76 6.02
ENOB= SNDR bits (2.8)
2.2.3.6 Offset and Gain Error
The transfer characteristic of an ADC is expected to be a straight line with uniform step width. However, the actual transfer step widths might not be uniform ideally. These non-ideal terms cause errors and non-linearity performance in ADCs.
Figure 2.6 (a) shows the offset error, which is defined as the horizontal deviation from the ideal position by a constant amount. The gain error (or scale factor error) describes the difference of slop between the ideal straight line and the actual transfer line, as shown in Figure 2.6 (b).
Input Output
actual ideal
( )a ( )b
Offset Error
Gain Error
actual ideal Output
Input
Figure 2.6 Illustrating (a) offset error and (b) gain error for a 3-bit A/D converter.
2.2.3.7 Differential Non-Linearity Error (DNL)
After both the offset and gain errors have been removed, each transfer step level might not be equal to 1 LSB (
2
ref N
=V ) ideally. The differential non-linearity error is
( )
, 11
step n
Width LSB
DNL n
LSB
= - (2.9)
An ADC is guaranteed not to have any missing codes if the minimum DNL error is larger than -1 LSB.
2.2.3.8 Integral Non-Linearity Error (INL)
The integral non-linearity error is defined as the deviation of the middle point of each transfer step form the ideal straight line. There are two ways to define the straight line. A common used definition is known as the endpoint straight line which is drawn through the end points of the first and last code transition. An alternative definition is to find the best-fit straight line such that the maximum INL is minimized [4]. The INL is also specified after both the offset and gain errors have been removed and can be expressed as
( )
( ), ( ),1
t n actual t n ideal
V V
INL n
LSB
= - (2.10)
Figure 2.7 shows the illustration of the DNL and INL.
Input Output
actual ideal
INL
1 LSB
1 DNL+ LSB
Figure 2.7 Illustrating the DNL and INL.
2.2.3.9 Sampling-Time Uncertainty (Aperture Jitter)
The sampling-time uncertainty is another significant issue that limits the performance of ADCs, which is also known as aperture jitter. Considering a sinusoidal wave input signal, V , with input frequency in f as below in
( )
sin 2( )
2
ref
in in
V t =V p f t (2.11)
Since the variance of V for a sinusoidal waveform is the largest at the zero crossing in point, we can find out the maximum slope by differentiating V with respect to time in and setting t=0, as shown below
max in
in ref
V f V
t p
D =
D (2.12) If Dt represents the sampling-time uncertainty, and if we want to keep DVin less than 1 LSB, we can find that
in in ref LSB
V p f V t V
D = D < (2.13)
In consequentially, we get the limit of the aperture jitter Dt of a N-bit ADC as follows
1 2
LSB
N
in ref in
t V
f V f
p p
D < = (2.14)
Figure 2.8 shows the concept of the aperture jitter [4].
2 Vref
2 Vref
-0
V
int
D t
V
inD
Sampling
Time
Figure 2.8 Aperture jitter.
2.2.3.10 Dynamic Range (DR)
The dynamic range is defined as the ratio between the maximum signal power for peak SNR and the minimum detectable signal power within a specified bandwidth.
With a sinusoidal input signal, we can measure the dynamic range by varying its amplitude to find the 0dB SNR and peak SNR positions, as shown in Figure 2.9. If the noise power is independent on the signal power, the dynamic range is equal to the SNR at full scale. However, generally the noise power increases as the signal power increases. Therefore, the actual peak SNR will be less than the dynamic range [3].
Dynamic Range
( )
SNR dB
( )
Vin dB
Peak SNR
0 dB
Figure 2.9 Dynamic range.