Distortion Analysis T
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- Bruce Hampton
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1 EE 435 Lecture 32 Spectral Performance Windowing Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Quantization Noise
2 . Review from last lecture. Distortion Analysis T A Χ m 2 N k 0 T S THEOREM: Consider a periodic signal with period T=1/f and sampling period T S =1/f S. If N P is an integer and x(t) is band limited to f MAX, then and 1 0 m h -1 Χ mn P for all k not defined above where Χ k N 1 k is the DFT of the sequence x kt S 0 N=number of samples, N P is the number of periods, and N 1 k 0 fmax 1 h = Int f N P Key Theorem central to Spectral Analysis that is widely used!!! and often abused
3 . Review from last lecture. Considerations for Spectral Characterization Tool Validation (MATLAB) FFT Length Importance of Satisfying Hypothesis Windowing
4 . Review from last lecture. Spectral Response (expressed in db) (Actually Stem plots but points connected in plotting program) Note Magnitude is Symmetric wrt f SAMPLE f AXIS f SIGNAL n 1 N P
5 . Review from last lecture. Input Waveform
6 . Review from last lecture. Example WLOG assume f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) ω 2πf SIG Consider N P =20.2 N=4096 Recall 20log 10 (0.5)=
7 . Review from last lecture. Input Waveform
8 . Review from last lecture. Spectral Response
9 Example WLOG assume f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) ω 2πf SIG Consider N P =20.01 N=4096 Deviation from hypothesis is.05% of the sampling window
10 Input Waveform
11 Input Waveform
12 Input Waveform
13 Input Waveform
14 Spectral Response with Non-Coherent Sampling (zoomed in around fundamental)
15 Fundamental will appear at position 1+Np = 21 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
16 k th harmonic will appear at position 1+k Np Columns 36 through
17 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!
18 Example WLOG assume f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) ω 2πf SIG Consider N P = N=4096 Deviation from hypothesis is.005% of the sampling window
19 Spectral Response with Non-coherent Sampling (zoomed in around fundamental)
20 Fundamental will appear at position 1+Np = 21 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
21 k th harmonic will appear at position 1+k Np Columns 36 through Columns 43 through Columns 50 through Columns 57 through
22 Observations Modest change in sampling window of 0.01 out of 20 periods (.005%) results in a small error in both fundamental and harmonic More importantly, substantial raise in the computational noise floor!!! (from over - 300dB to only -60dB) Errors at about the 10-bit level!
23 Spectral Response with Non-coherent sampling (zoomed in around fundamental)
24 Fundamental will appear at position 1+Np = 21 Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
25 k th harmonic will appear at position 1+k Np Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through 70
26 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!
27 Considerations for Spectral Characterization Tool Validation FFT Length Importance of Satisfying Hypothesis - NP is an integer - Band-limited excitation Windowing
28 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
29 Example If f SIG =50Hz and N P =20 N=512 N f 2f max SIGNAL N P f max < 640Hz
30 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)=
31 Effects of High-Frequency Spectral Components
32 Effects of High-Frequency Spectral Components
33 Effects of High-Frequency Spectral Components
34 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
35 Effects of High-Frequency Spectral Components f high =14fo Columns 36 through Columns 43 through Columns 50 through
36 Effects of High-Frequency Spectral Components
37 Effects of High-Frequency Spectral Components
38 Effects of High-Frequency Spectral Components (zoomed in around fundamental)
39 Effects of High-Frequency Spectral Components f 2f f Aliased components at f alias f sig thus position in 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
40 Effects of High-Frequency Spectral Components
41 Effects of High-Frequency Spectral Components
42 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
43 Considerations for Spectral Characterization Tool Validation FFT Length Importance of Satisfying Hypothesis - NP is an integer - Band-limited excitation Windowing
44 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
45 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 (also with appended zeros) Triangular Hamming Hanning Blackman
46 Rectangular Window Sometimes termed a boxcar window Uniform weight Can append zeros Without appending zeros equivalent to no window
47 Rectangular Window Assume f SIG =50Hz V IN sin( t) 0. 5sin( 2 t) ω 2πf SIG Consider N P =20.1 N=512
48 Rectangular Window
49 Spectral Response with Non-coherent sampling (zoomed in around fundamental)
50 Rectangular Window (with appended zeros)
51 Rectangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
52 Rectangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through Energy spread over several frequency components
53 Triangular Window
54 Triangular Window
55 Spectral Response with Non-Coherent Sampling and Windowing (zoomed in around fundamental)
56 Triangular Window
57 Triangular Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
58 Hamming Window
59 Hamming Window
60 Spectral Response with Non-Coherent Sampling and Windowing (zoomed in around fundamental)
61 Comparison with Rectangular Window
62 Hamming Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
63 Hanning Window
64 Hanning Window
65 Spectral Response with Non-Coherent Sampling and Windowing (zoomed in around fundamental)
66 Comparison with Rectangular Window
67 Hanning Window Columns 1 through Columns 8 through Columns 15 through Columns 22 through Columns 29 through
68 Comparison of 4 windows
69 Comparison of 4 windows
70 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
71 Comparison of 4 windows when sampling hypothesis are satisfied
72 Comparison of 4 windows
73 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
74 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
75 Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs
76 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
77 Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs
78 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
79 Quantization Effects 16,384 pts res = 4bits N P =25 20 msec
80 Quantization Effects 16,384 pts res = 4bits N P =25 20 msec
81 Quantization Effects 16,384 pts res = 4bits
82 Quantization Effects Simulation environment: N P =23 f SIG =50Hz V REF : -1V, 1V Res: will be varied N=2 n will be varied
83 Quantization Effects Res = 4 bits
84 Quantization Effects Res = 4 bits Axis of Symmetry
85 Quantization Effects Res = 4 bits Some components very small
86 Quantization Effects Res = 4 bits Set lower display limit at -120dB
87 Quantization Effects Res = 4 bits
88 Quantization Effects Res = 4 bits
89 Quantization Effects Res = 4 bits
90 Quantization Effects Res = 4 bits
91 Quantization Effects Res = 4 bits
92 Quantization Effects Res = 4 bits
93 Quantization Effects Res = 4 bits Fundamental
94 Quantization Effects Res = 10 bits
95 Quantization Effects Res = 10 bits
96 Quantization Effects Res = 10 bits
97 Quantization Effects Res = 10 bits
98 Quantization Effects Res = 10 bits
99 Quantization Effects Res = 10 bits
100 Quantization Effects Res = 10 bits
101 Quantization Effects Res = 10 bits
102 Quantization Effects Res = 10 bits
103 Quantization Effects Res = 10 bits
104 Quantization Effects Res = 10 bits
105 Quantization Effects Res = 10 bits
106 Quantization Effects Res = 10 bits
107 Quantization Effects Res = 10 bits Res 10 No. points 256 fsig= No. Periods Rectangular Window Columns 1 through Columns 6 through
108 Columns 11 through Columns 16 through Columns 21 through Columns 26 through Columns 31 through
109 Columns 36 through Columns 41 through Columns 46 through Columns 51 through Columns 56 through
110 Columns 61 through Columns 66 through Columns 71 through Columns 76 through Columns 81 through
111 Columns 86 through Columns 91 through
112 Res 10 No. points 4096 fsig= No. Periods Rectangular Window Columns 1 through Columns 6 through
113 Columns 11 through Columns 16 through Columns 21 through Columns 26 through Columns 31 through
114 Columns 36 through Columns 41 through Columns 46 through Columns 51 through Columns 56 through
115 Columns 61 through Columns 66 through Columns 71 through Columns 76 through Columns 81 through
116 Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs
117 Spectral Characteristics of DACs and ADCs
118 Spectral Characteristics of DAC t Periodic Input Signal Sampling Clock T SIG t Sampled Input Signal (showing time points where samples taken)
119 Spectral Characteristics of DAC T SIG Quantization Levels T PERIOD Quantized Sampled Input Signal (with zero-order sample and hold)
120 Spectral Characteristics of DAC T DFT WINDOW T PERIOD T SIG T CLOCK Sampling Clock T DFT CLOCK DFT Clock
121 Spectral Characteristics of DAC T DFT WINDOW T PERIOD T SIG T CLOCK Sampling Clock T DFT CLOCK DFT Clock
122 Spectral Characteristics of DAC Sampling Clock DFT Clock
123 Spectral Characteristics of DAC Sampled Quantized Signal (zoomed) DFT Clock Sampling Clock
124 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
125 DFT Simulation from Matlab n sam =
126 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
127 DFT Simulation from Matlab Expanded View n sam =
128 DFT Simulation from Matlab Expanded View n sam =
129 DFT Simulation from Matlab Expanded View n sam =
130 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
131 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
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 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
133 DFT Simulation from Matlab n sam =
134 DFT Simulation from Matlab Expanded View n sam =
135 DFT Simulation from Matlab Expanded View n sam =
136 DFT Simulation from Matlab Expanded View nsam =
137 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
138 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
139 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
140 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
141 DFT Simulation from Matlab n res =8
142 DFT Simulation from Matlab Expanded View n res =8
143 DFT Simulation from Matlab n res =8
144 DFT Simulation from Matlab Expanded View n res =8
145 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
146 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
147 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
148 DFT Simulation from Matlab n res =8
149 DFT Simulation from Matlab Expanded View n res =8
150 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
151 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
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 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
153 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
154 DFT Simulation from Matlab
155 DFT Simulation from Matlab
156 DFT Simulation from Matlab
157 DFT Simulation from Matlab
158 DFT Simulation from Matlab
159 DFT Simulation from Matlab
160 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
161 DFT Simulation from Matlab
162 DFT Simulation from Matlab
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 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
168 DFT Simulation from Matlab Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
169 DFT Simulation from Matlab Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
170 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
171 DFT Simulation from Matlab
172 DFT Simulation from Matlab
173 DFT Simulation from Matlab
174 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
175 DFT Simulation from Matlab
176 DFT Simulation from Matlab
177 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
178 DFT Simulation from Matlab Columns 36 through Columns 43 through Columns 50 through Columns 57 through Columns 64 through
179 DFT Simulation from Matlab Columns 71 through Columns 78 through Columns 85 through Columns 92 through Columns 99 through
180 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
181 End of Lecture 32
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