EE 230 Lecture 40. Data Converters. Amplitude Quantization. Quantization Noise

EE 230 Lecture 40 Data Converters Amplitude Quantization Quantization Noise

Review from Last Time: Time Quantization Typical ADC Environment

Review from Last Time: Time Quantization Analog Signal Reconstruction Smoothing filter removes some of the discontinuities in the output of the zero-order hold

Review from Last Time: Track and Hold Ф TH IN OUT

Review from Last Time: Amplitude Quantization Analog Signals at output of DAC are quantized Digital Signals at output of ADC are quantized t Desired Quantized

Review from Last Time: Amplitude Quantization Unwanted signals in the output of a system are called noise. Distortion Smooth nonlinearities Frequency attenuation Large Abrupt Nonlinearities Signals coming from other sources Movement of carriers in devices Interference from electrical coupling Interference from radiating sources Undesired outputs inherent in the data conversion process itself

Review from Last Time: Amplitude Quantization How big is the quantization noise characterized?

Amplitude Quantization Characterization of Quantization Noise Quantization noise is usually specified in terms of the rms value of the quantization error expressed relative to the LSB For convenience, consider the quantization noise for the following two waveforms that are as large as possible without without exceeding the input or output range of the data converter a) triangle or saw tooth b) sinusoidal

Characterization of Quantization Noise Saw tooth excitation Consider an ADC (flash) 111 110 101 100 011 010 001 000 OUT IN REF Assume first transition at REF /(2 (n+1) )

Characterization of Quantization Noise Saw tooth excitation QE t LSB

Characterization of Quantization Noise Saw tooth excitation Error waveform:

Characterization of Quantization Noise Saw tooth excitation Error waveform: Since x QE (t) is periodic, the QRMS can be obtained by integrating x QE2 (t) over one period 1 t 2 2 x () t dt QRMS = T t1 QE

Characterization of Quantization Noise Saw tooth excitation without loss of generality, can shift time axis so that QE is symmetric to the origin, thus T 2 t = 1 2 LSB x () t = t QE T t = T 2 for t 1 < t < t 2, can express QE as thus t2 T 2 1 1 () / QE T t1 T T / 2 T 2 2 LSB 2 = x t dt = t dt QRMS QRMS 2 2 3 T / 2 T / 2 2 t LSB LSB lsb 3 t dt 3 T T / 2 T 3 T / 2 12 = = =

Characterization of Quantization Noise Saw tooth excitation = QRMS lsb 12 QRMS 1 3 lsb Is this quantization noise signal small or large? Whether this is viewed as being large or small depends upon how the noise relates to the signal!

Signal to Noise Ratio Signal SNR= Noise Signal and Noise are generally expressed in terms of their RMS current or voltage values SNR= SIG-RMS NOISE-RMS Or, sometimes in terms of their Power values assumed driving resistive loads P SNR = P P SIG-RMS Thus SNR = P 2 2 SIG-RMS NOISE-RMS NOISE-RMS = SNR 2

Signal to Noise Ratio Signal and Noise often expressed in db But SNR =20log SNR db 10 =10log P-dB 10 P P SIG-RMS NOISE-RMS SIG NOISE 2 SIG-RMS SIG-RMS SNR =10log 20log =SNR 2 = NOISE-RMS NOISE-RMS P-dB 10 10 db SNR P-dB =SNR Often subscripts are dropped and will not cause a problem if in db SNR = SNR P-dB db =SNR db

Characterization of Quantization Noise Saw tooth excitation = QRMS lsb 12 What is the SNR of a data converter? SNR= SIG-RMS NOISE-RMS What is SIG-RMS?

Characterization of Quantization Noise Saw tooth excitation lsb SIG-RMS = SNR= QRMS 12 What is SIG-RMS? NOISE-RMS IN REF t = SIG RMS REF 12

Characterization of Quantization Noise Saw tooth excitation = but QRMS = LSB SNR = 1 2 lsb 12 n 2 REF 12 SNR= LSB 12 REF n = SIG RMS = REF LSB REF 12 Thus, the SNR for a data converter due to only the quantization noise is SNR = nlog 2 = 6. 02n db 10 or ( )

Characterization of Quantization Noise Saw tooth excitation SNR for a data converter due to only the quantization noise: SNR = 1 2 n SNR = 602. n db

Characterization of Quantization Noise Sinusoidal excitation IN Consider an ADC REF t Quantization noise is difficult to analytically characterize OUT 7 REF/8 3 REF/4 5 REF/8 REF/2 3 REF/8 REF/4 REF/8 0 t Still need RMS value of QE (t) QE t Will consider error in interpreted output LSB

Characterization of Quantization Noise Sinusoidal excitation Consider an ADC Will consider error in interpreted output QE LSB

Characterization of Quantization Noise Sinusoidal excitation Consider an ADC (clocked) Theorem: If n(t) is a random process, then 2 2 V σ + μ RMS provided that the RMS value is measured over a large interval where the parameters σ and μ are the standard deviation and the mean of <n(kt)> This theorem can thus be represented as V RMS t1 + T 1 L 2() 2 2 n t dt σ + μ T t1 L where T is the sampling interval and T L is a large interval

Characterization of Quantization Noise Sinusoidal excitation Consider an ADC The quantization noise samples of the ADC output are approximately uniformly distributed between in the interval [- LSB /2, LSB /2] <n(kt)> ~ U[- LSB /2, LSB /2] A random variable that is U[a,b] has distribution parameters μ and σ given by A + B B - A μ = σ = 2 12 thus, the random variable n(kt) has distribution parameters LSB μ = 0 σ = 12

Characterization of Quantization Noise Sinusoidal excitation μ = Consider an ADC n(kt) parameters 0 σ = LSB 12 It thus follows form the previous Theorem that 2 2 σ + μ Q-RMS 2 σ = σ Q-RMS Q-RMS LSB 12 Note that this is the same quantization noise voltage as was obtained with the triangular excitation!

Characterization of Quantization Noise Sinusoidal excitation Consider an ADC Q-RMS LSB 12 IN What is the SNR? REF REF () t = sin( ωt + θ) + SIG 2 2 1 REF REF SIG = = RMS 2 2 2 2 Observe that the RMS value for the sinusoidal signal differs from that of the triangular signal REF REF SIG = = RMS 12 2 3 REF t

Characterization of Quantization Noise Sinusoidal excitation Consider an ADC Q-RMS SIG RMS = LSB 12 REF 2 2 IN REF What is the SNR? t REF 2 2 3 REF REF REF SNR= = = 1. 225 = 1. 225 = 1. 225 2 n LSB 2 / 2 LSB LSB REF 12 REF REF SNR =20log 1.225 20log 1225. 602. n 176. db 10 10 n = = + /2 LSB REF n

Characterization of Quantization Noise Saw tooth excitation Sinusoidal excitation IN IN REF REF t t SNR =2 n SNR = 602. n db Q-RMS LSB 12 SNR = 1. 225 2 n SNR = 602. n + 176. db Although derived for an ADC, same expressions apply for DAC SNR for saw tooth and for triangle excitations are the same SNR for sinusoidal excitation larger than that for saw tooth by 1.76dB SNR will decrease if input is not full-scale Equivalent Number of Bits (ENOB) often given relative to quantization noise SNR db Remember quantization noise is inherent in an ideal data converter!

Recall: Unwanted signals in the output of a system are called noise. Distortion Smooth nonlinearities Frequency attenuation Large Abrupt Nonlinearities Signals coming from other sources Interference from electrical coupling Movement of carriers in devices Interference from radiating sources Undesired outputs inherent in the data conversion process itself

Equivalent Number of Bits (ENOB) Often other sources of noise are present in a data converter and often the noise or other nonidealities in the data converter result in errors that are larger than 1/3 LSB and in many cases even much larger than LSB ENOB often a figure of merit that is used to more effectively characterize the real resolution of a data converter if a full-scale sinusoidal input were applied SNR db,act : Actual Signal to Noise Ratio SNR = 602. n + 176. db,act EFF

Equivalent Number of Bits (ENOB) SNR = 602. n + 176. db,act EFF Often simply stated as n EFF = SNR -1.76 db,act 6.02 SNR - 1.76 ENOB = 6.02 Observe: ENOB is not dependent upon n

Equivalent Number of Bits (ENOB) ENOB = SNR - 1.76 6.02 Often distortion is also included in the noise category and ENOB is expressed as ENOB = SNDR - 1.76 6.02 where SNDR is the signal to (noise + distortion) ratio

Equivalent Number of Bits (ENOB) ENOB = SNR - 1.76 6.02 ENOB = SNDR - 1.76 6.02 These definitions of ENOB are based upon noise or noise and distortion Some other definitions of ENOB are used as well e.g. if one is only interested in distortion, an ENOB based upon distortion can be defined. ENOB is useful for determining whether the number of bits really being specified is really useful

Example: 8.5mV. The noise of a 12-bit ADC with V REF =5V was measured to be a) What is the quantization noise of this 12-bit ADC (in V RMS )? b) What is the ENOB? Quantization noise: QRMS QRMS 1 3 1 V 32 REF 12 lsb = 041. mv

Example: 8.5mV. The SNR of a 12-bit ADC with V REF =5V was measured to be a) What is the quantization noise of this 12-bit ADC (in V RMS )? b) What is the ENOB? Quantization noise: ENOB: QRMS 1 V 32 REF 12 = 041. mv SNR - 1.76 ENOB = V 6.02 REF 5V SNR = 2 2 1.77V 8.5mV SNR = 2 2 = = 208. 2 8.5mV 8.5mV ( SNR) ( ) SNR = 20log = 20log 208. 2 = 46. 4dB db 10 10 46.4-1.76 ENOB = =7.41 6.02 Note: In this application, an 8-bit ADC would give about the same SNR performance!

Engineering Issues for Using Data Converters 1. Inherent with Data Conversion Process Amplitude Quantization Time Quantization (Present even with Ideal Data Converters) 2. Nonideal Components Uneven steps Offsets Gain errors Response Time Noise (Present to some degree in all physical Data Converters) How do these issues ultimately impact performance?

End of Lecture 40