EE 505 Lecture 7. Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Clock Jitter Statistical Circuit Modeling

EE 505 Lecture 7 Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Clock Jitter Statistical Circuit Modeling

. Review from last lecture. MatLab comparison: 512 Samples with Standard Sweep Spectre Results MatLab Results

. Review from last lecture. MatLab comparison: 512 Samples with Strobe Period Sweep Spectre Results MatLab Results

. Review from last lecture. Comparison of 4 windows

. Review from last lecture. 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

. Review from last lecture. Post-processing Method of circumventing the coherent sampling problem Can also be used for addressing spectral purity problem for test signal generation x kt S N 1 k 0 Post-Processor Χ k N 1 k 0 Non-coherent Easily implemented in MATLAB Will be considered in the laboratory Removes fundamental from samples and replaces with coherent fundamental before taking DFT

Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs Clock Jitter

Quantization Effects (time and amplitude depicted) 16,384 pts res = 4bits N P =25 20 msec

Quantization Effects (time and amplitude depicted) 16,384 pts res = 4bits

Quantization Effects (time and amplitude depicted) Simulation environment: N P =23 f SIG =50Hz V REF : -1V, 1V Res: will be varied N=2 n will be varied

Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs Clock Jitter

Quantization Effects Res = 4 bits

Quantization Effects Res = 4 bits Axis of Symmetry

Quantization Effects Res = 4 bits

Quantization Effects Res = 4 bits Expect quantization noise effects to be uniformly distributed!!

Quantization Effects Res = 4 bits Note presence of odd-ordered harmonic terms!!

Res = 4 bits Why are there spectral components present in the quantization noise? Recall the uncorrelated assumption was good only for about 4 bits or more!

Quantization Effects Res = 10 bits Quantization noise is much more uniform

Quantization Effects Res = 10 bits

Quantization Effects Res = 10 bits Harmonic Components not Visible

. Review from last lecture. Quantization Effects Res = 10 bits Compared to the previous slide, it appears that the quantization noise has gone down why does this occur?

Quantization Effects Res = 10 bits Compared to the previous slide, it appears that the quantization noise has gone down even more why does this occur?

Quantization Effects Res = 10 bits Compared to the previous slides, it appears that the quantization noise has gone down even more why does this occur?

Quantization Effects Res = 10 bits

Quantization Effects Res = 10 bits Very small third harmonic component but does not extend above other noise terms

. Review from last lecture. Spectral Characterization Amplitude Quantization Does not introduce spectral components for n large Nearly uniformly distributed Decreases with increasing N

Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs Clock Jitter

Spectral Characteristics of DACs and ADCs

Spectral Characteristics of DAC t Periodic Input Signal Sampling Clock T SIG t Sampled Input Signal (showing time points where samples taken)

Spectral Characteristics of DAC T SIG Quantization Levels T PERIOD Quantized Sampled Input Signal (with zero-order sample and hold)

Spectral Characteristics of DAC T DFT WINDOW T PERIOD T SIG T CLOCK Sampling Clock T DFT CLOCK DFT Clock

Spectral Characteristics of DAC Sampling Clock DFT Clock

Spectral Characteristics of DAC Sampled Quantized Signal (zoomed) DFT Clock Sampling Clock

Spectral Characteristics of DAC Consider the following example f SIG =50 Hz f CL =500 Hz (DAC clock) f DFTCL =71.24K Hz n DFT =15 (coherent sampling) N P1 =23 (number of signal periods in DFT window) N P =1 n res =8 bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Matlab File: afft_quantization_dac_jan2017.m

DFT Simulation from Matlab n sam = 142.4696

DFT Simulation from Matlab Expanded View n sam = 142.4696 Width of this region is f CL Analogous to the overall DFT window when directly sampled but modestly asymmetric

DFT Simulation from Matlab Expanded View n sam = 142.4696

DAC Comparisons with Quantization Fundamental, second harmonic, and third harmonic N θ Nsam n A 1 A 2 A 3 32K 1 142.5 8 -.596-56.7-64.5 128K 1 569.9 8 -.596-56.7-64.45

Spectral Characteristics of DAC Consider the following example f SIG =50 Hz f CL =500 Hz (DAC clock) f DFTCL =71.24K Hz n DFT =18 (coherent sampling) N P1 =23 (number of signal periods in DFT window) N P =1 n res =8 bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Matlab File: afft_quantization_dac.m

DFT Simulation from Matlab n sam = 1139.75652

DFT Simulation from Matlab Expanded View n sam = 1139.75652

Spectral Characteristics of DAC Consider the following example f SIG =50 Hz f CL =500 Hz (DAC clock) f DFTCL =71.24K Hz n DFT =18 (coherent sampling) N P1 =23 (number of signal periods in DFT window) N P =1 n res =14bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Matlab File: afft_quantization_dac.m

Spectral Characteristics of DAC Consider the following example f SIG =50 Hz f CL =500 Hz (DAC clock) f DFTCL =71.24K Hz n DFT =18 (coherent sampling) N P1 =23 (number of signal periods in DFT window) N P =1 n res =16bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Matlab File: afft_quantization_dac.m

Spectral Characteristics of DAC Consider the following example f SIG =50 Hz f CL =497.8 Hz (DAC clock) f DFTCL =141.853K Hz n DFT =16 (not coherent sampling) N P1 =23 (number of signal periods in DFT window) N P =1 n res =16bits Xin(t) =.95sin(2πf SIG t) (-.4455dB) Matlab File: afft_quantization_dac.m

DFT Simulation from Matlab

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The spectral characteristics of DACs as affected by DAC clock rate, by DAC resolution, and by the length of the DFT window will be considered

Notation: T CL : Clock Period (=1/f CL ) T SIG : Signal Period (=1/f SIG ) T DFT : DFT period (=1/f DFT ) N SIG1 : Number of Signal Periods to establish periodicity with sampling clock N CL1 : Number of Clock Periods to establish periodicity with sampling clock N SIG : Number of Signal Periods in DFT window N CL : Number of Clock Periods in DFT window N P : Number of Periods in DFT window N SAM : Number of DFT samples in each clock period n: Length of DFT window in binary bits N: Length of DFT window n Q Number of binary bits of quantization

Graphical Interpretation of Signals for DAC Spectral Characterization T PERIOD T DFT Window T SIG T CLK T DFT Signal Clock Time Quantized Signal DFT CL In this example, N SIG1 =3, N CL1 =8, N P =2, N SIG =6, N CL =16

Consider the Following Examples Frequencies Number of Samples fsig fclk ffft NSIG1 NCLK1 NP NSIG NCL nfft N NSAM nq 1 a 50 500 3413.3 3 30 5 15 150 10 1024 6.8 12 1 b 50 500 3413.3 3 30 5 15 150 10 1024 6.8 24 1 c 50 500 3413.3 3 30 5 15 150 10 1024 6.8 6 1 d 50 500 54613.3 3 30 5 15 150 14 16384 109.2 12 1 e 50 500 54613.3 3 30 5 15 150 14 16384 109.2 24 1 f 50 500 54613.3 3 30 5 15 150 14 16384 109.2 6 2 a 50 500 853.3 3 30 5 15 150 8 256 1.7 18 2 b 50 500 3413.3 3 30 5 15 150 10 1024 6.8 18 2 c 50 500 13653.3 3 30 5 15 150 12 4096 27.3 18 2 d 50 500 54613.3 3 30 5 15 150 14 16384 109.2 18 2 e 50 500 218453.3 3 30 5 15 150 16 65536 436.9 18 3 a 50 500 13653.3 3 30 5 15 150 12 4096 27.3 4 3 b 50 500 13653.3 3 30 5 15 150 12 4096 27.3 6 3 c 50 500 13653.3 3 30 5 15 150 12 4096 27.3 8 3 d 50 500 13653.3 3 30 5 15 150 12 4096 27.3 10 3 e 50 500 13653.3 3 30 5 15 150 12 4096 27.3 12 3 f 50 500 13653.3 3 30 5 15 150 12 4096 27.3 14 3 g 50 500 13653.3 3 30 5 15 150 12 4096 27.3 16 4 a 50 550 204800.0 1 11 1 1 11 12 4096 372.4 16 4 b 50 1550 10778.9 1 11 19 19 589 12 4096 7.0 16 4 c 50 500 13653.3 1 31 19 15 150 12 4096 27.3 16 4 d 50 5000 22755.6 3 30 5 9 900 12 4096 4.6 16 4 e 50 5000 22755.6 3 300 3 9 900 12 4096 4.6 16 4 e 50 5000 68266.7 3 300 1 3 300 12 4096 13.7 16

Quantization Effects N Q =6, 12 and 24 DFT = 1024

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 12 No. points 1024 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 6.83 Rectangular Window Pyyt = Columns 1 through 7-68.1563-62.9858-120.0000-53.8578-110.4355-53.2167-120.0000 Columns 8 through 14-71.8295-112.6608-50.4921-120.0000-36.9792-117.2313-93.0830 Columns 15 through 21-119.8393-0.5943-120.0000-51.4435-115.8566-70.0418-120.0000

Columns 22 through 28-55.2740-115.5905-54.0499-120.0000-61.5572-120.0000-64.3935 Columns 29 through 35-120.0000-53.5785-80.8146-50.7997-120.0000-73.8544-106.7187 Columns 36 through 42-49.2731-120.0000-42.6700-113.4803-83.4962-120.0000-38.0130 Columns 43 through 49-120.0000-52.1961-117.9058-69.8846-120.0000-57.0939-118.1488 Columns 50 through 56-54.1607-120.0000-60.0797-120.0000-65.8099-120.0000-53.2026 Columns 57 through 63-113.0039-47.7940-120.0000-76.2962-82.6398-47.6681-120.0000

Columns 64 through 70-45.8177-106.9044-78.9700-120.0000-44.3694-119.2650-52.7931 Columns 71 through 77-119.2863-67.0634-120.0000-58.7508-119.6253-54.1937-119.4593 Columns 78 through 84-58.5220-120.0000-67.2805-119.1209-52.7161-120.0000-43.6734 Columns 85 through 91-120.0000-79.4967-108.1915-45.4485-120.0000-47.9278-86.1269 Columns 92 through 98-75.9120-120.0000-48.2738-111.5480-53.2627-120.0000-65.6057 Columns 99 through 105-120.0000-60.2949-120.0000-54.1497-117.8663-56.8453-120.0000

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 24 No. points 1024 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 6.83 Rectangular Window Pyyt = Columns 1 through 7-120.0000-62.9769-120.0000-53.8590-120.0000-53.2154-120.0000 Columns 8 through 14-71.8639-120.0000-50.4930-120.0000-36.9796-120.0000-93.1002 Columns 15 through 21-120.0000-0.5945-120.0000-51.4444-120.0000-70.1050-120.0000 Columns 22 through 28-55.2722-120.0000-54.0510-120.0000-61.5510-120.0000-64.3799

Columns 29 through 35-120.0000-53.5796-120.0000-50.7989-120.0000-73.8798-120.0000 Columns 36 through 42-49.2739-120.0000-42.6705-120.0000-83.5140-120.0000-38.0130 Columns 43 through 49-120.0000-52.1971-120.0000-68.5096-120.0000-57.0915-120.0000 Columns 50 through 56-54.1619-120.0000-60.0753-120.0000-65.7865-120.0000-53.2037 Columns 57 through 63-120.0000-47.7935-120.0000-76.3173-120.0000-47.6689-120.0000 Columns 64 through 70-45.8184-120.0000-78.9892-120.0000-44.3692-120.0000-52.7941

Columns 71 through 77-120.0000-67.0165-120.0000-58.7474-120.0000-54.1949-120.0000 Columns 78 through 84-58.5188-120.0000-67.2252-120.0000-52.7172-120.0000-43.6732 Columns 85 through 91-120.0000-79.5155-120.0000-45.4491-120.0000-47.9286-120.0000 Columns 92 through 98-75.9335-120.0000-48.2733-120.0000-53.2638-120.0000-65.5843 Columns 99 through 105-120.0000-60.2903-120.0000-54.1509-120.0000-56.8430-120.0000 Columns 106 through 112-68.7296-120.0000-52.1000-120.0000-36.6381-120.0000-84.3923

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 6 No. points 1024 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 6.83 Rectangular Window Pyyt = Columns 1 through 7-32.0327-63.5202-84.8031-53.9597-74.3119-53.4184-90.1917 Columns 8 through 14-70.6391-76.5372-50.6042-96.7252-37.1128-81.1077-92.5357 Columns 15 through 21-83.7157-0.7348-92.4277-51.5529-79.7330-67.8750-90.8641 Columns 22 through 28-55.4986-79.4669-54.1512-84.7153-61.9728-92.1910-65.1368

Columns 29 through 35-84.5558-53.6813-44.6910-50.9834-89.3517-72.9967-70.5951 Columns 36 through 42-49.3883-102.1167-42.7976-77.3567-82.9263-85.6839-38.1619 Columns 43 through 49-88.1742-52.3033-81.7822-54.1377-91.3896-57.3481-82.0252 Columns 50 through 56-54.2616-84.2571-60.4194-92.0212-67.0068-83.9688-53.3066 Columns 57 through 63-76.8803-47.9632-88.3136-75.5982-46.5162-47.7871-112.9329 Columns 64 through 70-45.9404-70.7808-78.3491-87.2319-44.5283-83.1414-52.8984

Columns 71 through 77-83.1627-69.4679-91.7825-59.0446-83.5017-54.2945-83.3357 Columns 78 through 84-58.8094-91.7341-70.1569-82.9973-52.8217-83.9530-43.8308 Columns 85 through 91-87.0314-78.8852-72.0679-45.5718-110.4069-48.0462-50.0033 Columns 92 through 98-75.1969-88.4755-48.4449-75.4244-53.3665-84.0746-66.7081 Columns 99 through 105-92.0526-60.6437-84.3479-54.2506-81.7427-57.0948-91.3230 Columns 106 through 112-57.3951-81.5372-52.2065-88.8073-36.7857-85.4320-83.8094

Comparison of DAC resolution levels 12-bit resolution DFT = 1024 6-bit resolution 24-bit resolution

Quantization Effects N Q =6, 12 and 14 DFT = 16,384

Rectangular Window Pyyt = Columns 1 through 7-68.1643-71.8575-120.0000-97.8684-120.0000-77.3089-120.0000 Columns 8 through 14-78.1289-120.0000-84.2363-120.0000-91.2471-120.0000-76.2515 Columns 15 through 21-120.0000-0.5890-120.0000-117.4015-120.0000-61.7284-120.0000 Columns 22 through 28-76.8397-120.0000-89.8220-120.0000-85.6720-120.0000-77.9413 Columns 29 through 35-120.0000-74.9649-80.7871-100.2208-120.0000-69.7526-120.0000

Columns 36 through 42-73.3986-120.0000-95.9035-120.0000-79.3134-120.0000-78.2395 Columns 43 through 49-120.0000-82.7262-120.0000-90.0258-120.0000-75.5146-120.0000 Columns 50 through 56-60.9480-120.0000-107.8171-120.0000-68.1314-120.0000-77.3059 Columns 57 through 63-120.0000-88.4445-120.0000-87.0619-82.6269-77.6710-120.0000 Columns 64 through 70-72.0737-120.0000-103.2629-120.0000-66.6269-120.0000-74.5792 Columns 71 through 77-120.0000-94.1627-120.0000-81.0912-120.0000-78.2763-120.0000

Columns 78 through 84-81.1067-120.0000-94.1473-120.0000-74.5888-120.0000-66.5906 Columns 85 through 91-120.0000-103.2957-120.0000-72.0436-120.0000-77.6682-86.1466 Columns 92 through 98-87.0745-120.0000-88.4319-120.0000-77.3097-120.0000-68.1747 Columns 99 through 105-120.0000-107.7636-120.0000-61.0239-120.0000-75.5070-120.0000 Columns 106 through 110-86.6053-120.0000-82.7119-120.0000-78.2402

Res 24 No. points 16384 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 109.23 Rectangular Window Pyyt = Columns 1 through 7-120.0000-71.8582-120.0000-97.8939-120.0000-77.3077-120.0000 Columns 8 through 14-78.1301-120.0000-84.2318-120.0000-91.1966-120.0000-76.2525 Columns 15 through 21-120.0000-0.5892-120.0000-117.4187-120.0000-61.7285-120.0000 Columns 22 through 28-76.8408-120.0000-89.7993-120.0000-85.6657-120.0000-77.9424

Columns 29 through 35-120.0000-74.9641-120.0000-100.2421-120.0000-69.7533-120.0000 Columns 36 through 42-73.3994-120.0000-95.9379-120.0000-79.3117-120.0000-78.2407 Columns 43 through 49-120.0000-82.7229-120.0000-92.6592-120.0000-75.5155-120.0000 Columns 50 through 56-60.9484-120.0000-107.8349-120.0000-68.1312-120.0000-77.3070 Columns 57 through 63-120.0000-88.4310-120.0000-87.0530-120.0000-77.6722-120.0000 Columns 64 through 70-72.0733-120.0000-103.2819-120.0000-66.6274-120.0000-74.5801

Columns 71 through 77-120.0000-94.2246-120.0000-81.0888-120.0000-78.2775-120.0000 Columns 78 through 84-81.1043-120.0000-94.2097-120.0000-74.5897-120.0000-66.5911 Columns 85 through 91-120.0000-103.3147-120.0000-72.0431-120.0000-77.6693-120.0000 Columns 92 through 98-87.0656-120.0000-88.4185-120.0000-77.3107-120.0000-68.1745 Columns 99 through 105-120.0000-107.7813-120.0000-61.0243-120.0000-75.5080-120.0000 Columns 106 through 110-92.6647-120.0000-82.7086-120.0000-78.2414

Res 6 No. points 16384 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 109.23 Rectangular Window Pyyt = Columns 1 through 7-32.0407-71.9762-108.6491-97.0037-100.4389-77.5108-111.2274 Columns 8 through 14-78.2302-101.6042-84.5790-120.0000-93.8523-103.7705-76.3588 Columns 15 through 21-115.4515-0.7295-107.5027-116.8544-106.0217-61.8770-114.9359 Columns 22 through 28-76.9452-100.6561-90.9847-120.0000-86.0896-95.4491-78.0432

Columns 29 through 35-112.4573-75.1491-44.6635-99.5160-108.8855-69.8755-116.2764 Columns 36 through 42-73.5137-108.0918-94.7109-107.5841-79.5374-109.7041-78.3404 Columns 43 through 49-105.2173-83.0170-116.5842-52.7519-105.7895-75.6240-115.8410 Columns 50 through 56-61.0817-108.3689-107.2486-103.4880-68.2896-114.2808-77.4099 Columns 57 through 63-94.9896-89.1814-120.0000-87.5956-46.5033-77.7738-113.4590 Columns 64 through 70-72.2437-98.3943-102.6459-108.8046-66.7547-116.1157-74.6913

Columns 71 through 77-107.1739-92.0373-112.4162-81.3438-107.7781-78.3771-107.7581 Columns 78 through 84-81.3596-112.4559-92.0056-107.1636-74.7008-116.1137-66.7185 Columns 85 through 91-108.8023-102.6793-98.4649-72.2135-113.4673-77.7710-50.0230 Columns 92 through 98-87.6096-120.0000-89.1664-94.9127-77.4136-114.2741-68.3330 Columns 99 through 105-103.4571-107.1947-108.3746-61.1575-115.8441-75.6165-105.8047 Columns 106 through 110-59.8339-116.5472-83.0023-105.2445-78.3411

Comparison of DAC resolution levels DFT = 16, 384 6-bit resolution 24-bit resolution 12-bit resolution

DFT Length Effects DFT = 8,10,12, 14, and 16 bits

Res 18 No. points 256 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 1.71 Rectangular Window Pyyt = Columns 1 through 7-120.0000-47.7622-120.0000-34.2868-120.0000-42.8432-120.0000 Columns 8 through 14-42.4097-120.0000-49.4943-120.0000-37.6688-120.0000-43.8076 Columns 15 through 21-120.0000-0.5819-120.0000-48.8053-120.0000-46.1425-120.0000 Columns 22 through 28-39.5711-120.0000-40.8180-120.0000-45.2879-120.0000-49.0879

Columns 29 through 35-120.0000-26.2821-120.0000-43.9606-120.0000-34.6434-120.0000 Columns 36 through 42-49.3816-120.0000-43.7127-120.0000-42.1857-120.0000-36.9259 Columns 43 through 49-120.0000-47.1928-120.0000-48.2414-120.0000-30.1647-120.0000 Columns 50 through 56-43.3359-120.0000-40.8298-120.0000-49.5423-120.0000-39.9035 Columns 57 through 63-120.0000-43.5270-120.0000-26.8711-120.0000-48.4487-120.0000 Columns 64 through 70-46.8705-120.0000-37.9405-120.0000-41.7860-120.0000-44.2825

Res 18 No. points 1024 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 6.83 Rectangular Window Pyyt = Columns 1 through 7-120.0000-62.9769-120.0000-53.8590-120.0000-53.2154-120.0000 Columns 8 through 14-71.8637-120.0000-50.4930-120.0000-36.9796-120.0000-93.1001 Columns 15 through 21-120.0000-0.5945-120.0000-51.4444-120.0000-70.1047-120.0000 Columns 22 through 28-55.2723-120.0000-54.0510-120.0000-61.5511-120.0000-64.3800

Columns 29 through 35-120.0000-53.5796-120.0000-50.7989-120.0000-73.8797-120.0000 Columns 36 through 42-49.2739-120.0000-42.6705-120.0000-83.5140-120.0000-38.0130 Columns 43 through 49-120.0000-52.1971-120.0000-68.5158-120.0000-57.0915-120.0000 Columns 50 through 56-54.1619-120.0000-60.0753-120.0000-65.7866-120.0000-53.2037 Columns 57 through 63-120.0000-47.7935-120.0000-76.3172-120.0000-47.6689-120.0000 Columns 64 through 70-45.8184-120.0000-78.9891-120.0000-44.3692-120.0000-52.7941

Res 18 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-120.0000-72.6097-120.0000-59.1123-120.0000-66.9167-120.0000 Columns 8 through 14-64.8141-120.0000-65.2075-120.0000-73.3392-120.0000-71.3611 Columns 15 through 21-120.0000-0.5884-120.0000-67.9000-120.0000-60.6922-120.0000 Columns 22 through 28-68.6157-120.0000-73.6210-120.0000-69.4421-120.0000-58.4188

Columns 29 through 35-120.0000-68.0323-120.0000-49.1201-120.0000-70.8144-120.0000 Columns 36 through 42-73.4795-120.0000-66.5144-120.0000-63.7630-120.0000-67.3444 Columns 43 through 49-120.0000-55.5743-120.0000-72.2712-120.0000-72.9025-120.0000 Columns 50 through 56-61.6305-120.0000-66.3715-120.0000-65.6628-120.0000-63.6417 Columns 57 through 63-120.0000-73.1508-120.0000-71.8365-120.0000-49.2898-120.0000 Columns 64 through 70-67.6771-120.0000-62.3927-120.0000-67.6638-120.0000-73.5754

Res 18 No. points 16384 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 109.23 Rectangular Window Pyyt = Columns 1 through 7-120.0000-71.8582-120.0000-97.8938-120.0000-77.3077-120.0000 Columns 8 through 14-78.1301-120.0000-84.2318-120.0000-91.1968-120.0000-76.2525 Columns 15 through 21-120.0000-0.5892-120.0000-117.4186-120.0000-61.7285-120.0000 Columns 22 through 28-76.8408-120.0000-89.7995-120.0000-85.6657-120.0000-77.9424

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 18 No. points 65536 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 436.91 Rectangular Window Pyyt = Columns 1 through 7-120.0000-73.5038-120.0000-94.8920-120.0000-90.9985-120.0000 Columns 8 through 14-97.7033-120.0000-90.4462-120.0000-94.2570-120.0000-79.6113 Columns 15 through 21-120.0000-0.5889-120.0000-95.4453-120.0000-91.4307-120.0000 Columns 22 through 28-97.6566-120.0000-89.7549-120.0000-93.5290-120.0000-83.1981

Columns 29 through 35-120.0000-73.2712-120.0000-95.9255-120.0000-91.7555-120.0000 Columns 36 through 42-97.5628-120.0000-88.8976-120.0000-92.6930-120.0000-85.7407 Columns 43 through 49-120.0000-79.1390-120.0000-96.5032-120.0000-91.9818-120.0000 Columns 50 through 56-97.4211-120.0000-87.8332-120.0000-91.7284-120.0000-87.7021 Columns 57 through 63-120.0000-82.4858-120.0000-96.6925-120.0000-92.1154-120.0000 Columns 64 through 70-97.2300-120.0000-86.4975-120.0000-90.6065-120.0000-89.2883

DFT Length Effects

Comparison of DAC Resolution Levels Resolutions: = 4,6,8,10,12, 14, and 16 bits

Res 4 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-19.9968-79.0080-76.4899-58.8098-99.7385-66.7313-84.0072 Columns 8 through 14-64.6189-96.3196-64.8105-83.4783-74.8303-88.3521-69.5563 Columns 15 through 21-71.5456-0.3380-76.9482-67.7195-81.1346-60.4804-87.6422 Columns 22 through 28-68.0172-86.5089-74.7460-87.1411-68.7215-85.0196-58.1996 Columns 29 through 35

Res 6 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-32.0380-73.8391-88.5311-59.2410-111.7797-67.0712-96.0484 Columns 8 through 14-64.9665-108.3608-65.3153-95.5195-73.8343-100.3933-71.1342 Columns 15 through 21-83.5868-0.7287-88.9894-68.0556-93.1758-60.8410-99.6834 Columns 22 through 28-68.6780-98.5501-74.0462-99.1823-69.4766-97.0608-58.5660 Columns 29 through 35

Res 8 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-44.0792-72.3617-100.5723-59.1224-120.0000-66.9203-108.0896 Columns 8 through 14-64.8182-120.0000-65.2230-107.5607-73.2589-112.4345-71.4638 Columns 15 through 21-95.6280-0.5957-101.0306-67.9033-105.2170-60.6973-111.7246 Columns 22 through 28-68.6427-110.5913-73.5576-111.2235-69.4763-109.1020-58.4243 Columns 29 through 35

Res 10 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-56.1204-72.8046-112.6135-59.1148-120.0000-66.9239-120.0000 Columns 8 through 14-64.8209-120.0000-65.2062-119.6019-73.4080-120.0000-71.2971 Columns 15 through 21-107.6692-0.5931-113.0718-67.9075-117.2582-60.6984-120.0000 Columns 22 through 28-68.6059-120.0000-73.6773-120.0000-69.4272-120.0000-58.4247

Res 12 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-68.1616-72.6331-120.0000-59.1118-120.0000-66.9168-120.0000 Columns 8 through 14-64.8141-120.0000-65.2065-120.0000-73.3470-120.0000-71.3523 Columns 15 through 21-119.7104-0.5882-120.0000-67.9001-120.0000-60.6922-120.0000 Columns 22 through 28-68.6136-120.0000-73.6272-120.0000-69.4395-120.0000-58.4188 Columns 29 through 35

Columns 29 through 35 Res 14 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-80.2028-72.6053-120.0000-59.1123-120.0000-66.9166-120.0000 Columns 8 through 14-64.8140-120.0000-65.2076-120.0000-73.3377-120.0000-71.3627 Columns 15 through 21-120.0000-0.5884-120.0000-67.8999-120.0000-60.6921-120.0000 Columns 22 through 28-68.6159-120.0000-73.6198-120.0000-69.4425-120.0000-58.4187

Res 16 No. points 4096 fsig = 50.00 No.DFT Periods 5.00 No Sig Periods 15.00 fcl/fsig 10.00 Nsamp = 27.31 Rectangular Window Pyyt = Columns 1 through 7-98.2593-72.6097-120.0000-59.1123-120.0000-66.9167-120.0000 Columns 8 through 14-64.8141-120.0000-65.2075-120.0000-73.3392-120.0000-71.3611 Columns 15 through 21-120.0000-0.5884-120.0000-67.9000-120.0000-60.6922-120.0000 Columns 22 through 28-68.6157-120.0000-73.6210-120.0000-69.4421-120.0000-58.4188

Comparison of DAC Resolution Levels

DAC Comparisons with Quantization N θ Nsam n A 1 A 2 A 3 32K 1 142.5 8 -.596-56.7-64.5 128K 1 569.9 8 -.596-56.7-64.45 1024 1 6.8 6 -.735-44.7-54.1 1024 1 6.8 12 -.594-80.8-69.6 1024 1 6.8 24 -.594-120 -68.5 16K 1 109.2 6 -.729-44.7-52.7 16K 1 109.2 12 -.589-80.8-90 16K 1 109.2 14 -.589-120 -92.7 256 1 1.7 18 -.589-120 -48.2 1024 1 6.8 18 -.595-120 -68.5 4048 1 27.3 18 -.588-120 -72.3 16K 1 436.9 18 -.589-120 -96.5

Res 12 No. points 16384 fsig = 50.00 No.DFT Periods 1.00 No Sig Periods 1.00 fcl/fsig 11.00 Nsamp = 1489.45 Rectangular Window Pyyt = Columns 1 through 7-67.0548-0.5647-74.8202-80.2380-76.8384-87.7740-85.8216 Columns 8 through 14-87.4527-91.8275-93.9240-20.5665-120.0000-22.1465-83.4794 Columns 15 through 21-85.8696-80.6925-89.1920-87.9067-84.3703-89.8040-89.3743 Columns 22 through 28-27.0127-120.0000-27.7956-84.7890-86.7910-81.4391-89.1242

Res 16 No. points 4096 fsig = 50.00 No.DFT Periods 19.00 No Sig Periods 19.00 fcl/fsig 11.00 Nsamp = 19.60 Rectangular Window Pyyt = Columns 1 through 7-76.5382-90.5279-75.2699-88.1107-75.0808-85.1251-74.7552 Columns 8 through 14-82.2078-74.2777-79.3331-73.6136-76.3251-72.7191-72.9179 Columns 15 through 21-71.5113-68.6004-69.8366-61.8629-67.3812-0.5633-63.2788 Columns 22 through 28-60.6082-52.9838-66.0354-57.1836-68.9816-66.3803-70.9147

Res 16 No. points 4096 fsig = 50.00 No.DFT Periods 19.00 No Sig Periods 19.00 fcl/fsig 31.00 Nsamp = 6.95 Rectangular Window Pyyt = Columns 1 through 7-107.4387-82.4247-81.4377-74.3745-80.8993-93.5561-83.2968 Columns 8 through 14-85.8707-47.1114-78.2880-91.7183-83.7001-89.2039-73.4847 Columns 15 through 21-73.1349-89.0683-83.6903-91.8215-78.4438-0.4606-85.6974 Columns 22 through 28-83.2682-93.6131-80.9876-74.8192-81.1939-82.3711-94.2516

Res 16 No. points 4096 fsig = 50.00 No.DFT Periods 3.00 No Sig Periods 9.00 fcl/fsig 100.00 Nsamp = 4.55 Rectangular Window Pyyt = Columns 1 through 7-91.6204-90.5217-120.0000-89.6572-120.0000-86.7242-120.0000 Columns 8 through 14-82.6204-120.0000-0.4470-120.0000-79.5167-120.0000-86.7243 Columns 15 through 21-120.0000-89.0053-120.0000-90.5737-120.0000-90.6480-120.0000 Columns 22 through 28-89.6632-120.0000-87.9065-120.0000-82.6205-120.0000-72.6534 Columns 29 through 35

Res 16 No. points 4096 fsig = 50.00 No.DFT Periods 1.00 No Sig Periods 3.00 fcl/fsig 100.00 Nsamp = 13.65 Rectangular Window Pyyt = Columns 1 through 7-91.6204-89.6627-120.0000-0.4470-120.0000-88.9986-120.0000 Columns 8 through 14-89.6541-120.0000-72.6847-120.0000-88.9963-120.0000-90.1484 Columns 15 through 21-120.0000-72.6733-120.0000-88.1328-119.8900-90.1410-120.0000 Columns 22 through 28-78.5504-120.0000-88.1286-117.2233-90.4653-120.0000-78.7969 Columns 29 through 35

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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

Duty Cycle Effects on Spectral Performance of DACS File: DAC Quantization with RTZ.m

Consider N P =1 N SIG =11 N CL =70 f sig =50 n res =10 Thus, f CLK =f SIG (N CL /N SIG )=318Hz The fft spectrum should be nominally symmetric around f CLK /2=159Hz so will get only the fundamental, second harmonic, and third harmonic in the fundamental frequency half-period which occurs at fft coefficient number 36 and the clock frequency will be at fft coefficient number 71 (and thus the fundamental will appear at fft coefficient numbers 11+1=12 and 71-11=60) The relationship between fft coefficient number and frequency is given by n-1 f= f NSIG SIG or by n=1+f N f SIG SIG

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 10 No. points 16384 fsig = 50.00 No.DFT Periods 1.00 No Sig Periods 11.00 fcl/fsig 6.36 Nsamp = 234.06 DutyCycle = 1.0 Rectangular Window Pyyt = Columns 1 through 8-56.7666-72.6329-89.0180-65.9223-84.2996-73.2991-67.3277-63.0000 Columns 9 through 16-78.0844-73.9060-80.2415-0.8009-71.7226-76.1473-77.2781-64.7624 Columns 17 through 24-83.8268-72.4855-81.0600-71.7684-84.3311-72.4003-100.5411-75.2036

Columns 25 through 32-104.6890-87.5996-86.8260-81.1797-95.8796-74.4617-94.9546-71.5626 Columns 33 through 40-79.4929-82.0122-108.8848-86.4078-108.8871-73.6154-80.9806-75.0515 Columns 41 through 48-97.3320-78.9052-99.3163-80.8769-91.3537-74.6389-110.0719-77.5449 Columns 49 through 56-107.4100-75.6450-92.3523-72.3248-90.2704-78.7130-94.4099-64.8687 Columns 57 through 64-89.3611-75.9678-85.3927-15.3935-95.8308-75.1766-95.9254-63.5195 Columns 65 through 72-87.8618-73.6845-108.7233-68.9982-119.8229-71.7477-120.0000-71.7563

Columns 73 through 80-119.9494-68.7360-109.5559-73.6204-89.4074-63.5185-97.8093-74.8683 Columns 81 through 88-98.2726-18.1394-88.4301-76.0204-92.7995-65.1698-98.4133-75.7393 Columns 89 through 96-94.8485-72.0469-97.3332-76.9476-112.8736-76.5337-116.8212-79.5798 Columns 97 through 104-98.2141-81.0207-106.8397-76.9870-105.5319-79.2621-89.5668-79.9400 Columns 105 through 110-118.9287-86.4077-117.6606-76.3449-90.0484-82.8245

Magnitude of Fundamental 0.950 2nd Harmonic 0.000 in db -0.4-220.0 Res 10 No. points 16384 fsig = 50.00 No.DFT Periods 1.00 No Sig Periods 11.0 fcl/fsig 6.36 Nsamp = 234.06 DutyCycle = 0.5 Rectangular Window Columns 1 through 8-64.9875-75.2613-95.1326-71.2094-90.2852-76.6156-73.2632-69.5014 Columns 9 through 16-83.9643-77.8162-86.0866-6.5546-77.4246-78.1520-82.8739-67.8450 Columns 17 through 24-89.2754-77.9987-86.4061-73.3492-89.4464-72.4374-105.4623-75.4661 Columns 25 through 32-109.3846-77.1183-91.3829-74.6853-100.0604-83.0708-98.8928-75.4658

Columns 33 through 40-83.0515-77.2072-113.3805-73.0081-111.6998-82.3913-83.3412-72.6823 Columns 41 through 48-99.2516-77.3944-100.3706-69.8376-92.1989-78.5482-111.3365-67.6419 Columns 49 through 56-106.9480-72.1848-91.2497-64.1067-88.1852-72.9575-91.1348-57.9429 Columns 57 through 64-85.1895-73.1598-79.8865-9.1712-88.8113-71.4550-86.7115-58.0188 Columns 65 through 72-76.4621-69.7368-93.6087-64.6782-98.9534-67.5523-64.9561-67.2996 Columns 73 through 80-98.9315-63.4010-94.7247-69.9035-77.9584-58.0242-89.2150-71.7656

Columns 81 through 88-91.2239-11.9238-82.9849-72.9920-88.6074-58.0323-95.6827-74.9840 Columns 89 through 96-92.7477-63.9572-95.8693-76.5352-112.3992-67.3668-114.7685-74.7990 Columns 97 through 104-99.0492-69.8650-109.1443-75.9390-107.4431-70.6650-92.0134-75.6831 Columns 105 through 110-120.0000-72.9982-117.8073-77.0787-93.6013-72.8613

Summary of Duty Cycle Effects Duty Cycle does dot introduce harmonic distortion Duty Cycle reduction reduces signal levels thus degrades SNR

Number of Samples/Period One Sample per Period Multiple Samples per Period Many authors use a data acquisition system and select one sample/period Spectrum analyzer will generally measure continuous-time effects

Number of Samples/Period Nonlinear Settling Glitch Glitch Typical DAC Response

Number of Samples/Period Incorrect Settling Typical DAC Response

Number of Samples/Period Incorrect Settling Settling error can be multiple LSB at Nyquist Rate Multiple LSB settling error does not cause distortion if settling is linear Glitches are a significant contributor to spectral distortion (at high frequencies)

Typical SFDR Plot From: Y. Cong and R. L. Geiger, "A 1.5-v 14-bit 100-MS/s Self-Calibrated DAC," IEEE J. of Solid State Circuits, December 2003, vol. 38, no. 12, pp. 2051-2060.

Spectral Characterization of Data Converters Distortion Analysis Time Quantization Effects of DACs of ADCs Amplitude Quantization Effects of DACs of ADCs Clock Jitter

Effects of Jitter on Spectral Performance

Model of Jitter T S Ideal Clock Jittery Clock t Jk Assume t Jk are uncorrelated uniformly distributed random variables θ θ tjk U - T S, TS 2 2 Note: there can also be jitter in the ideal clock or there may be no ideal clock so zero crossings may be modeled as a random walk or a sum of a random walk and uniform jitter. Analysis more complicated in these cases.

Model of Jitter Assume t Jk are uncorrelated uniformly distributed random variables θ θ tjk U - T S, TS 2 2 Consider θ=.01,.001,.0001,.00001 Observe: If T S is a 100MHz clock, then T S =10nsec and θ=.0001 corresponds to 1psec (±0.5psec) of symmetric jitter

Statistical Characterization of Electronic Components and Circuits Recall: Almost all data converter structures work perfectly if components are ideal Major challenges in data converter design Parasitic Resistances and Capacitances Nonlinearity in components Statistical variation in components and circuits Model uncertainties Power supply variability

Consider a flash ADC V REF V IN R R R R Thermometer to Binary Decoder n X OUT Resistor values and offset voltages of Comparators are all random variables at design level Variations of these RVs affect the break point and thus the yield R

Consider Current-Steering DAC V DD I 1 I 1 /2 I 1 /4 I 1 /8 b 3 b 2 b 1 b 0 R F V OUT Ideally n -1 b i OUT 1 F n-i i=0 2 V = -I R

Consider Current-Steering DAC V DD V DD M REF M n-1 M 1 M 0 I 1 I 1 /2 I 1 /4 I 1 /8 b 3 b 2 b 1 b 0 R F V R I n-1 I 1 I 0 V OUT I REF b n-1 M (n-1)s M 1S b0 b 1 M 0S Basic Implementation of Current Sources Ideally Actually I μc W V -V 2 OX k k R Tp 2 L k L k=l0 k-1 W k=2 W0 I μc W V -V 2 k OXk k k R Tpk 2 L k I k is a random variables and is a function of the model parameters μ k, C OXk, W k, L k, and V Tpk μ k, C OXk, W k, L k, and V Tpk are all random variables

Recall from previous lecture How important is statistical analysis? Example: 7-bit FLASH ADC with R-string DAC V REF V IN Assume R-string is ideal, V REF =1V and V OS for each comparator must be at most +/- ½ LSB Case 1 Standard deviation is 5mV Case 2 P COMP = 0.565 Y =3.2 10 ADC -32 R R R R Thermometer to Binary Decoder n X OUT Standard deviation is 1mV P COMP 0.999904 Y =0.988 ADC R Statistics play a key role in the performance and consequently yield of a data converter

End of Lecture 7