EE 435. Lecture 32. Spectral Performance Windowing
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1 EE 435 Lecture 32 Spectral Performance Windowing
2 . Review from last lecture. Distortion Analysis T 0 T S THEOREM?: If N P is an integer and x(t) is band limited to f MAX, then 2 Am Χ mnp 1 0 m h N and Χ k for all k not defined above where f = 1/T, Χ k N k f N f MAX = 2 N 1 is the DFT of the sequence x kt 0 S P, and h = Int f MAX f N 1 k 0
3 . Review from last lecture. Considerations for Spectral Characterization Tool Validation FFT Length Importance of Satisfying Hypothesis - NP is an integer - Band-limited excitation Windowing
4 . Review from last lecture. Observations Modest change in sampling window of out of 20 periods (.0005%) results in a small error in both fundamental and harmonic More importantly, substantial raise in the computational noise floor!!! (from over - 300dB to only -80dB) Errors at about the 13-bit level!
5 . Review from last lecture. Effects of High-Frequency Spectral Components
6 . Review from last lecture. Observations Aliasing will occur if the band-limited part of the hypothesis for using the DFT is not satisfied Modest aliasing will cause high frequency components that may or may not appear at a harmonic frequency More egregious aliasing can introduce components near or on top of fundamental and lower-order harmonics Important to avoid aliasing if the DFT is used for spectral characterization
7 Considerations for Spectral Characterization Tool Validation FFT Length Importance of Satisfying Hypothesis - NP is an integer - Band-limited excitation Windowing
8 Are there any strategies to address the problem of requiring precisely an integral number of periods to use the FFT? Windowing is sometimes used Windowing is sometimes misused
9 Windowing Windowing is the weighting of the time domain function to maintain continuity at the end points of the sample window Well-studied window functions: Rectangular Triangular Hamming Hanning Blackman
10 Rectangular Window Sometimes termed a boxcar window Uniform weight Can append zeros Without appending zeros equivalent to no window
11 Rectangular Window Assume f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) ω 2πf SIG Consider N P =20.1 N=512
12
13 Rectangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
14 Rectangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through Energy spread over several frequency components
15 Triangular Window
16 Triangular Window
17
18 Triangular Window
19 Triangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
20 Hamming Window
21 Hamming Window
22
23 Comparison with Rectangular Window
24 Hamming Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
25 Hanning Window
26 Hanning Window
27
28 Comparison with Rectangular Window
29 Hanning Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
30 Comparison of 4 windows
31 Comparison of 4 windows
32 Preliminary Observations about Windows Provide separation of spectral components Energy can be accumulated around spectral components Simple to apply Some windows work much better than others But windows do not provide dramatic improvement and
33 Comparison of 4 windows when sampling hypothesis are satisfied
34 Comparison of 4 windows
35 Preliminary Observations about Windows Provide separation of spectral components Energy can be accumulated around spectral components Simple to apply Some windows work much better than others But windows do not provide dramatic improvement and can significantly degrade performance if sampling hypothesis are met
36 Issues of Concern for Spectral Analysis An integral number of periods is critical for spectral analysis Not easy to satisfy this requirement in the laboratory Windowing can help but can hurt as well Out of band energy can be reflected back into bands of interest Characterization of CAD tool environment is essential Spectral Characterization of high-resolution data converters requires particularly critical consideration to avoid simulations or measurements from masking real performance
37 End of Lecture 30
38 EE 435 Lecture 31 Quantization Noise Absolute and Relative Accuracy
39 . Review from last lecture. Distortion Analysis T f MAX, then 2 Am N and Χ k 0 T S THEOREM: If N P is an integer and x(t) is band limited to where f = 1/T, and Χ k 1 0 m h -1 Χ mn N k P for all k not defined above 1 0 is the DFT of the sequence x kt S f N f MAX = 2 N P N 1 k 0
40 . Review from last lecture. Observations Modest change in sampling window of 0.01 out of 20 periods (.05%) still results in a modest error in both fundamental and harmonic More importantly, substantial raise in the computational noise floor!!! (from over - 300dB to only -40dB) Errors at about the 6-bit level!
41 . Review from last lecture. FFT Examples Recall the theorem that provided for the relationship between the DFT terms and the Fourier Series Coefficients required 1. The sampling window be an integral number of periods 2. N 2f f max SIGNAL N P
42 . Review from last lecture. Example If f SIG =50Hz and N P =20 N=512 N 2f f max SIGNAL N P f max < 640Hz
43 . Review from last lecture. Example Consider N P =20 N=512 If f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) 0. 5sin( 14 t) ω 2πf SIG (i.e. a component at 700 Hz which violates the band limit requirement) Recall 20log 10 (0.5)=
44 . Review from last lecture. Effects of High-Frequency Spectral Components
45 . Review from last lecture. Effects of High-Frequency Spectral Components
46 . Review from last lecture. Effects of High-Frequency Spectral Components
47 . Review from last lecture. Effects of High-Frequency Spectral Components Columns 1 through 7 f high =14fo Columns 8 through Columns 15 through Columns 22 through Columns 29 through
48 . Review from last lecture. Effects of High-Frequency Spectral Components f high =14fo Columns 36 through Columns 43 through Columns 50 through
49 . Review from last lecture. Effects of High-Frequency Spectral Components f 2f f Aliased components at f alias f sig thus positionin sequence 1 Columns 225 through f sig alias 11. 6f sample Columns 232 through Columns 239 through Columns 246 through N sig p f f alias sig
50 . Review from last lecture. Effects of High-Frequency Spectral Components
51 . Review from last lecture. Effects of High-Frequency Spectral Components
52 . Review from last lecture. Effects of High-Frequency Spectral Components
53 . Review from last lecture. Effects of High-Frequency Spectral Components
54 . Review from last lecture. Effects of High-Frequency Spectral Components
55 . Review from last lecture. Observations Aliasing will occur if the band-limited part of the hypothesis for using the DFT is not satisfied Modest aliasing will cause high frequency components that may or may not appear at a harmonic frequency More egregious aliasing can introduce components near or on top of fundamental and lower-order harmonics Important to avoid aliasing if the DFT is used for spectral characterization
56 . Review from last lecture. Windowing Windowing is the weighting of the time domain function to maintain continuity at the end points of the sample window Well-studied window functions: Rectangular Triangular Hamming Hanning Blackman
57 . Review from last lecture. Comparison of 4 windows
58 . Review from last lecture. Comparison of 4 windows when sampling hypothesis are satisfied
59 . Review from last lecture. Issues of Concern for Spectral Analysis An integral number of periods is critical for spectral analysis Not easy to satisfy this requirement in the laboratory Windowing can help but can hurt as well Out of band energy can be reflected back into bands of interest Characterization of CAD tool environment is essential Spectral Characterization of high-resolution data converters requires particularly critical consideration to avoid simulations or measurements from masking real performance
60 Spectral Characterization Distortion Analysis Time Quantization Effects Spectral Characteristic of DAC Time and Amplitude Quantization
61 Will leave the issues of time-quantization and DAC characterization to the student These concepts are investigated in the following slides Concepts are important but time limitations preclude spending more time on these topics in this course
62 Skip to Next Yellow Slide Few comments from slides
63 Spectral Characterization Distortion Analysis Time Quantization Effects Spectral Characteristic of DAC Time and Amplitude Quantization
64 Quantization Effects on Spectral Performance and Noise Floor in DFT Assume the effective clock rate (for either an ADC or a DAC) is arbitrarily fast Without Loss of Generality it will be assumed that f SIG =50Hz Index on DFT will be listed in terms of frequency (rather than index number) Matlab File: afft_quantization.m
65 Quantization Effects 16,384 pts res = 4bits N P =25 20 msec
66 Quantization Effects 16,384 pts res = 4bits N P =25 20 msec
67 Quantization Effects 16,384 pts res = 4bits
68 Quantization Effects Simulation environment: N P =23 f SIG =50Hz V REF : -1V, 1V Res: will be varied N=2 n will be varied
69 Quantization Effects Res = 4 bits
70 Quantization Effects Res = 4 bits Axis of Symmetry
71 Quantization Effects Res = 4 bits Some components very small
72 Quantization Effects Res = 4 bits Set lower display limit at -120dB
73 Quantization Effects Res = 4 bits
74 Quantization Effects Res = 4 bits
75 Quantization Effects Res = 4 bits
76 Quantization Effects Res = 4 bits
77 Quantization Effects Res = 4 bits
78 Quantization Effects Res = 4 bits
79 Quantization Effects Res = 4 bits Fundamental
80 Quantization Effects Res = 10 bits
81 Quantization Effects Res = 10 bits
82 Quantization Effects Res = 10 bits
83 Quantization Effects Res = 10 bits
84 Quantization Effects Res = 10 bits
85 Quantization Effects Res = 10 bits
86 Quantization Effects Res = 10 bits
87 Quantization Effects Res = 10 bits
88 Quantization Effects Res = 10 bits
89 Quantization Effects Res = 10 bits
90 Quantization Effects Res = 10 bits
91 Quantization Effects Res = 10 bits
92 Quantization Effects Res = 10 bits
93 Quantization Effects Res = 10 bits Res 10 No. points 256 fsig= No. Periods Rectangular Window Columns 1 through Columns 6 through
94 Columns 11 through Columns 16 through Columns 21 through Columns 26 through Columns 31 through
95 Columns 36 through Columns 41 through Columns 46 through Columns 51 through Columns 56 through
96 Columns 61 through Columns 66 through Columns 71 through Columns 76 through Columns 81 through
97 Columns 86 through Columns 91 through
98 Res 10 No. points 4096 fsig= No. Periods Rectangular Window Columns 1 through Columns 6 through
99 Columns 11 through Columns 16 through Columns 21 through Columns 26 through Columns 31 through
100 Columns 36 through Columns 41 through Columns 46 through Columns 51 through Columns 56 through
101 Columns 61 through Columns 66 through Columns 71 through Columns 76 through Columns 81 through
102 Quantization Effects Res = 10 bits With Vin=2v pp
103 With Vin=1*.99 and Vos=.25LSB
104 With Vin = pp
105 With Vin=1*.99 and Vos=.35LSB
106 Res 10 No. points 4096 fsig= No. Periods Tstep e-004 Magnitude of Fundamental nd Harmonic Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
107 Res = 10 bits Quantization Effects
108 Res = 10 bits Quantization Effects
109 Res = 10 bits Quantization Effects
110 Res = 10 bits Quantization Effects
111 Res = 5 bits Quantization Effects
112 Res = 4 bits Quantization Effects
113 Res = 4 bits Quantization Effects
114
115 Quantization Effects 16,384 pts res = 4bits
116 16,384 pts res = 4bits Quantization Effects
117 Res = 10 bits Quantization Effects
118 Spectral Characterization Distortion Analysis Time Quantization Effects Spectral Characteristic of DAC Time and Amplitude Quantization
119 Spectral Characteristics of DACs and ADCs
120 Spectral Characteristics of DAC t Periodic Input Signal Sampling Clock T SIG t Sampled Input Signal (showing time points where samples taken)
121 Spectral Characteristics of DAC T SIG Quantization Levels T PERIOD Quantized Sampled Input Signal (with zero-order sample and hold)
122 Spectral Characteristics of DAC T DFT WINDOW T PERIOD T SIG T CLOCK Sampling Clock T DFT CLOCK DFT Clock
123 Spectral Characteristics of DAC T DFT WINDOW T PERIOD T SIG T CLOCK Sampling Clock T DFT CLOCK DFT Clock
124 Spectral Characteristics of DAC Sampling Clock DFT Clock
125 Spectral Characteristics of DAC Sampled Quantized Signal (zoomed) DFT Clock Sampling Clock
126 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =230 k 2 =23 N P =1 n res =8bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Thus N P1 =23 θ SR =5 f CL /f SIG =10 Matlab File: afft_quantization_dac.m
127 DFT Simulation from Matlab n sam =
128 DFT Simulation from Matlab Expanded View n sam = Width of this region is f CL Analogous to the overall DFT window when directly sampled but modestly asymmetric
129 DFT Simulation from Matlab Expanded View n sam =
130 DFT Simulation from Matlab Expanded View n sam =
131 DFT Simulation from Matlab Expanded View n sam =
132 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) N=32768 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
133 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) N=32768 Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
134 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) N=32768 Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
135 DFT Simulation from Matlab n sam =
136 DFT Simulation from Matlab Expanded View n sam =
137 DFT Simulation from Matlab Expanded View n sam =
138 DFT Simulation from Matlab Expanded View nsam =
139 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) N= Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
140 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) Columns 36 through 42 N= Columns 43 through Columns 50 through Columns 57 through Columns 64 through
141 f SIG =50Hz, k 1 =23, k 2 =23, N P =1, n res =8bits Xin(t) =sin(2πf SIG t) N= Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
142 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =50 k 2 =5 N P =2 n res =8bits Xin(t) = =.95sin(2πf SIG t) (-.4455dB) Thus N P1 =5 θ SR =5 N P2 =10
143 DFT Simulation from Matlab n res =8
144 DFT Simulation from Matlab Expanded View n res =8
145 DFT Simulation from Matlab n res =8
146 DFT Simulation from Matlab Expanded View n res =8
147 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) N= Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
148 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) Columns 36 through 42 N= Columns 43 through Columns 50 through Columns 57 through Columns 64 through
149 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) N= Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
150 DFT Simulation from Matlab n res =8
151 DFT Simulation from Matlab Expanded View n res =8
152 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) N=1024 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
153 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) N=1024 Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
154 f SIG =50Hz, k 1 =50, k 2 =5, N P =2, n res =8bits, Xin(t) =sin(2πf SIG t) N=1024 Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
155 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =11 k 2 =1 N P =2 n res =12bits Xin(t) = =.95sin(2πf SIG t) (-.4455dB) Thus N P1 =1 θ SR =11 N P2 =2
156 DFT Simulation from Matlab
157 DFT Simulation from Matlab
158 DFT Simulation from Matlab
159 DFT Simulation from Matlab
160 DFT Simulation from Matlab
161 DFT Simulation from Matlab
162 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =230 k 2 =23 N P =1 n res =12bits Xin(t) = =.95sin(2πf SIG t) (-.4455dB) Thus N P1 =23 θ SR =10 N P2 =23
163 DFT Simulation from Matlab
164 DFT Simulation from Matlab
165 DFT Simulation from Matlab
166 DFT Simulation from Matlab
167 DFT Simulation from Matlab
168 DFT Simulation from Matlab
169 DFT Simulation from Matlab f SIG =50Hz k 1 =230 k 2 =23 N P =1 n res =12bits Xin(t) = =.95sin(2πfSIGt) (-.4455dB) N P1 =23 θ SR =10 N P2 =23 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
170 DFT Simulation from Matlab Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
171 DFT Simulation from Matlab Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
172 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =230 k 2 =23.1 N P =1 n res =12bits Xin(t) = =.95sin(2πf SIG t) (-.4455dB) Thus N P1 =23.1 θ SR =9.957 N P2 =23.1
173 DFT Simulation from Matlab
174 DFT Simulation from Matlab
175 DFT Simulation from Matlab
176 Spectral Characteristics of DAC Consider the following example f SIG =50Hz k 1 =230 k 2 =23 N P =1 n res =12bits Xin(t) =.88sin(2πf SIG t)+0.1sin(2πf SIG t) (-1.11db fundamental, -20dB 2 nd harmonic) Thus N P1 =23 θ SR =10 N P2 =23
177 DFT Simulation from Matlab
178 DFT Simulation from Matlab
179 Columns 1 through 7 DFT Simulation from Matlab f SIG =50Hz k 1 =230 k 2 =23 N P =1 n res =12bits Xin(t) =.88sin(2πf SIG t)+0.1sin(2πf SIG t) (-1.11db fundamental, -20dB 2nd harmonic) N P1 =23 θ SR =10 N P2 = Columns 8 through Columns 15 through Columns 22 through Columns 29 through
180 DFT Simulation from Matlab Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
181 DFT Simulation from Matlab Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
182 Skip from Previous Yellow Slide
183 Summary of time and amplitude quantization assessment Time and amplitude quantization do not introduce harmonic distortion Time and amplitude quantization do increase the noise floor
184 Quantization Noise DACs and ADCs generally quantize both amplitude and time If converting a continuous-time signal (ADC) or generating a desired continuoustime signal (DAC) these quantizations cause a difference in time and amplitude from the desired signal First a few comments about Noise
185 Noise We will define Noise to be the difference between the actual output and the desired output of a system Types of noise: Random noise due to movement of electrons in electronic circuits Interfering signals generated by other systems Interfering signals generated by a circuit or system itself Error signals associated with imperfect signal processing algorithms or circuits
186 Noise We will define Noise to be the difference between the actual output and the desired output of a system All of these types of noise are present in data converters and are of concern when designing most data converters Can not eliminate any of these noise types but with careful design can manage their effects to certain levels Noise (in particular the random noise) is often the major factor limiting the ultimate performance potential of many if not most data converters
187 Noise We will define Noise to be the difference between the actual output and the desired output of a system Types of noise: Random noise due to movement of electrons in electronic circuits Interfering signals generated by other systems Interfering signals generated by a circuit or system itself Error signals associated with imperfect signal processing algorithms or circuits Quantization noise is a significant component of this noise in ADCs and DACs and is present even if the ADC or DAC is ideal
188 Quantization Noise in ADC (same concepts apply to DACs) Consider an Ideal ADC with first transition point at 0.5X LSB X IN ADC n X OUT X REF If the input is a low frequency sawtooth waveform of period T that goes from 0 to X REF, the error signal in the time domain will be: ε Q.5 X LSB T 1 2T 1 3T 1 4T 1 T t -.5 X LSB where T 1 =T/2 n This time-domain waveform is termed the Quantization Noise for the ADC with a sawtooth (or triangular) input
189 Quantization Noise in ADC ε Q.5 X LSB T 1 2T 1 3T 1 4T 1 T t -.5 X LSB For large n, this periodic waveform behaves much like a random noise source that is uncorrelated with the input and can be characterized by its RMS value which can be obtained by integrating over any interval of length T 1. For notational convenience, shift the waveform by T 1 /2 units E RMS 1 T T /2 1 1 T /2 1 2 Q t dt
190 Quantization Noise in ADC ε Q ε Q.5 X LSB T 1 2T 1 3T 1 4T 1 T t -0.5T 1.5 X LSB 0.5T 1 t -.5 X LSB -.5 X LSB E RMS 1 T T /2 1 1 T /2 1 2 Q t dt In this interval, ε Q can be expressed as XLSB Q t T1 t
191 Quantization Noise in ADC E RMS 1 T T /2 1 1 T /2 1 2 Q t dt -0.5T 1 ε Q.5 X LSB 0.5T 1 t T /2 1 1 X E LSB RMS - t T T E X 1 T / t T /2 RMS LSB 3 T 3 1 -T / dt -.5 X LSB XLSB Q t T1 t E RMS X LSB 12
192 Quantization Noise in ADC E RMS X LSB 12 The signal to quantization noise ratio (SNR) can now be determined. Since the input signal is a sawtooth waveform of period T and amplitude X REF, it follows by the same analysis that it has an RMS value of X RMS Thus the SNR is given by or, in db, X REF 12 X SNR = RMS X RMS 2 E X RMS LSB SNR =20 n log2 =6.02n db Note: db subscript often neglected when not concerned about confusion n
193 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? X IN X REF t SNR =20 n log2 =6.02n
194 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? X REF X IN t Time and amplitude quantization points
195 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? X REF X IN t X QIN Time and Amplitude Quantized Waveform t
196 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? X REF X IN t X QIN ε Q t X LSB t Error waveform
197 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? ε Q X LSB t Appears to be highly uncorrelated with input even though deterministic Mathematical expression for ε Q very messy Excursions exceed X LSB (but will be smaller and bounded by ± X LSB /2 for lower frequency signal/frequency clock ratios) For lower frequency inputs and higher resolution, at any time, errors are approximately uniformly distributed between X LSB /2 and X LSB /2 Analytical form for ε QRMS essentially impossible to obtain from ε Q (t)
198 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? 0.5X LSB ε Q -0.5X LSB t For low f SIG /f CL ratios, bounded by ±XLB and at any point in time, behaves almost as if a uniformly distributed random variable ε Q ~ U[-0.5X LSB, 0.5X LSB ]
199 Recall: Quantization Noise in ADC If the random variable f is uniformly distributed in the interval [A,B] f : U[A,B] then the mean and standard deviation of f are given by A+B B-A μ f = σ= f 2 12 Theorem: If n(t) is a random process, then for large T, t 1+T V = n t dt = σ +μ 1 RMS n n T t
200 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? 0.5X LSB ε Q -0.5X LSB t εq ~ U[-0.5XLSB, 0.5XLSB] A+B μ = Q 2 0 B-A X σ= LSB f t 1+T V = n t dt = σ +μ 1 RMS n n T t V = RMS Q X LSB Note this is the same RMS noise that was present with a triangular input 12
201 Quantization Noise in ADC How does the SNR change if the input is a sinusoid that goes from 0 to X REF centered at X REF /2? ε Q 0.5X LSB -0.5X LSB X V LSB RMS = 12 But XREF 1 V INRMS= 2 2 Thus obtain XREF 2 2 n 3 SNR = = 2 XLSB 2 12 Finally, in db, n 3 SNR db = 20log 2 =6.02 n t
202 ENOB based upon Quantization Noise Solving for n, obtain SNR = 6.02 n ENOB = SNRdB Note: could have used the SNR db for a triangle input and would have obtained the expression SNR db ENOB = 6.02 But the earlier expression is more widely used when specifying the ENOB based upon the noise level present in a data converter
203 ENOB based upon Quantization Noise For very low resolution levels, the assumption that the quantization noise is uncorrelated with the signal is not valid and the ENOB expression will cause a modest error n 4 3 from van de Plassche (p13) corr SNR 2-2+ π 2 Table values in db Res (n) SNR corr SNR Almost no difference for n 3 SNR = 6.02 n +1.76
204 End of Lecture 32
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