EE 230 Lecture 43 Data Converters
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: Characterization of Quantization Noise Sinusoidal excitation Consider an ADC (cloced) 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(t)> 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
Review from Last Time: Characterization of Quantization Noise Saw tooth excitation Sinusoidal excitation X IN X IN X REF X REF t t SNR =2 n SNR = 602. n db XQ-RMS X 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!
Review from Last Time: 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
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?
Nonideal Transfer Characteristics Uneven Steps X OUT Nonideal Transfer Characteristics of DAC Ideal Transfer Characteristic Line X IN
Nonideal Transfer Characteristics Uneven Steps X OUT Nonideal Transfer Characteristics of DAC X OUT Nonideal Transfer Characteristics of DAC Ideal Transfer Characteristic Line Ideal Transfer Characteristic Line X IN X IN Actual transfer characteristics can vary considerably from one device to another
Nonideal Transfer Characteristics Uneven Steps X OUT Nonideal Transfer Characteristics of DAC Ideal Transfer Characteristic Line X IN This is termed a nonlinearity in the data converter Linearity metrics (specifications) include, DNL, THD and SFDR
Characterization of Nonlinearities X OUT Nonideal Transfer Characteristics of DAC End Point End Point X IN End points are the outputs at the two extreme Boolean inputs
Characterization of Nonlinearities End point fit line
Characterization of Nonlinearities Linearity metrics: DNL THD SFDR Integral Nonlinearity () Measure of worst-case deviation from linear Define the at any input code by: =X ( ) -X ( ) OUT FIT
Integral Nonlinearity () Linearity metrics: DNL THD SFDR Define the by: { } = max 1 N
Integral Nonlinearity () { } = max 1 N Often expressed in LSB: where LSB = X LSBF ( ) ( ) X N -X 1 OUT OUT X = LSBF N-1 Linearity metrics: DNL THD SFDR
Integral Nonlinearity () { } = max 1 N Often expressed in LSB: where LSB = X LSBF ( ) ( ) X N -X 1 OUT OUT X = LSBF N-1 Linearity metrics: DNL THD SFDR but ( ) ( ) X N -X 1 X X = LSBF N-1 2 OUT OUT REF n LSB n 2 X REF
Characterization of Nonlinearities Linearity metrics: DNL THD SFDR Differential Nonlinearity (DNL) Measure of worst-case resolving capabilities Define the DNL at any input code by: ( ) ( ) DNL =X ()-X -1 -X X ()-X -1 -X OUT OUT LSBF OUT OUT LSB
Differential Nonlinearity (DNL) Linearity metrics: DNL THD SFDR Often expressed in LSB DNL LSB ( ) DNL X ()-X -1 -X OUT OUT LSB { } DNL = max DNL LSBF 1< N DNL = DNL 2 LSB X n DNL X REF
Characterization of Nonlinearities IN OUT Linearity metrics: DNL THD SFDR LK IN OUT LK Linearity Metrics for ADC and DAC are Analogous to Each Other
Integral Nonlinearity () Linearity metrics: DNL THD SFDR
Integral Nonlinearity () Linearity metrics: DNL THD SFDR
Integral Nonlinearity () Linearity metrics: DNL THD SFDR =X ( ) -X ( ) TRAN FIT () TRAN X ()=X 1 + FIT TRAN N-2 N-2 ( ) ( 1) -1 X N-1 -X TRAN
Integral Nonlinearity () Linearity metrics: DNL THD SFDR =X ( ) -X ( ) TRAN FIT { } = max 1 N LSB = X LSBF LSB n 2 X REF
Differential Nonlinearity (DNL) Linearity metrics: DNL THD SFDR DNL X ()-X -1 -X ( ) TRANS TRANS LSB { } DNL = max DNL 1< N
Equivalent Number of Bits -ENOB (based upon linearity) Generally expect to be less than ½ LSB If larger than ½ LSB, effective resolution is less than specified resolution
Equivalent Number of Bits -ENOB (based upon linearity) If ν is the in LSB log ENOB = n-1 log 10 10 ν 2 ν 0.5 1 2 4 8 16 res n n-1 n-2 n-3 n-4 n-5
Spectral Characterization Linearity metrics: DNL THD SFDR X IN n DAC X OUT C LK IN M ( ) X=Xsinωt+θ If nonlinearities present, X OUT given by OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ = 2
Spectral Characterization IN M ( ) X=Xsinωt+θ OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ = 2 THD A, >1 are all spectral distortion components Generally only first few terms are large enough to represent significant distortion = = 2 A A 1 2 2 THD db = 10log 2 A = 2 10 2 A1 1 1 SFDR= db 10 max { A } max{ A } 1< A SFDR =20log 1< A
Spectral Characterization IN M ( ) X=Xsinωt+θ OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ = 2 Generally X M is chosen nearly full-scale and input is offset by X REF /2 X X 2 2 ( ) REF REF X= + ε sinωt+θ IN Direct measurement of A terms not feasible A generally calculated from a large number of samples of X OUT (t)
Spectral Characterization IN M ( ) X=Xsinωt+θ OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ Key theorem useful for spectral characterization = 2 Theorem: If a periodic signal x(t) with period T=1/f is band-limited to frequency hf and if the signal is sampled N times over an integral number of periods, N P, then where ( ) 2 A = X mn +1 m P N N 1 X = 1 the sampling period. ( ) for 0 m h-1 x T S is the DFT of the sampled sequence ( ) N 1 = 1 where T S is T N T= S N P
Spectral Characterization IN M ( ) X=Xsinωt+θ OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ Key theorem useful for spectral characterization = 2 Theorem: If a periodic signal x(t) with period T=1/f is band-limited to frequency hf and if the signal is sampled N times over an integral number of periods, N P, then where ( ) 2 A = X mn +1 m P N N 1 X = 1 the sampling period. ( ) for 0 m h-1 x T S is the DFT of the sampled sequence ( ) N 1 = 1 where T S is This theorem is usually not stated although widely used Often this theorem is misunderstood or misused If hypothesis not exactly satisfied, major problems with trying to use this theorem
Spectral Characterization IN M ( ) X=Xsinωt+θ OUT 0 1 1 ( γ ) + ( + γ ) X =A +Asin ωt+θ+ A sin ωt+θ Key theorem useful for spectral characterization = 2 Theorem: If a periodic signal x(t) with period T=1/f is band-limited to frequency hf and if the signal is sampled N times over an integral number of periods, N P, then where ( ) 2 A = X mn +1 m P N N 1 X = 1 the sampling period. ( ) for 0 m h-1 x T S is the DFT of the sampled sequence ( ) N 1 = 1 where T S is This theorem is usually not stated although widely used Often this theorem is misunderstood or misused If hypothesis not exactly satisfied, major problems with trying to use this theorem
Spectral Characterization Key theorem useful for spectral characterization Theorem: If a periodic signal x(t) with period T=1/f is band-limited to frequency hf and if the signal is sampled N times over an integral number of periods, N P, then where ( ) 2 A = X mn +1 m P N N 1 X = 1 the sampling period. ( ) for 0 m h-1 x T S is the DFT of the sampled sequence ( ) N 1 = 1 where T S is Usually N p is a prime number (e.g. 11, 21, 29, 31) If N is a power of 2, the Fast Fourier Transform (FFT) is a computationally efficient method for calculating the DFT Often N=4096, 65,536, FFT is available in Matlab and as subroutines for C++
Spectral Characterization Key theorem useful for spectral characterization Theorem: If a periodic signal x(t) with period T=1/f is band-limited to frequency hf and if the signal is sampled N times over an integral number of periods, N P, then where ( ) 2 A = X mn +1 m P N N 1 X = 1 the sampling period. ( ) for 0 m h-1 x T S is the DFT of the sampled sequence ( ) N 1 = 1 where T S is A 0, A 1, A 2, A 3, are the magnitudes of the DFT elements X(0), X(N P +1), X(2N P +1), X(3N P +1), respectively
Spectral Characterization
Spectral Characterization Χ( ) 1 2 0 3 4 N P +1 2N P +1 3N P +1 4N P +1