1. Theoretical Analysis of CMOS Computational Circuits for Analog Signal Processing Prof. dr. ing. Cosmin Radu POPA March 2015

2. 1. Introduction

3. Advantages of analog computation: - Low-power operation - High speed - High accuracy - Small silicon areas Research objectives: - Improvement of circuits’ accuracies - Low-voltage low-power operations - Reduction of circuits’ complexity - Increasing the number of developed circuit functions

4. 1. Introduction 2. Fundamental CMOS computational structures 2.1. Squaring circuits 2.2. Multiplier/divider circuits 2.3. Euclidean distance circuits 2.4. Active resistor structures 3. Multifunctional structures 4. CMOS function synthesizers 4.1. General function synthesizers 4.2. Exponential function synthesizers 4.3. Gaussian function synthesizers 4.4. Sinh and tanh function synthesizers 5. Conclusions OUTLINE

5. 2. Fundamental CMOS computational structures

6. 2.1. Squaring circuits

7. 2.1. Squaring circuits Voltage-input circuits

8. Squaring circuit (I)

9. Squaring circuit (I) – general schematic 2.1. Squaring circuits

10. Squaring circuit (I) – first realization 2.1. Squaring circuits

11. Squaring circuit (I) – second realization 2.1. Squaring circuits

12. 2.1. Squaring circuits Voltage-input circuits Squaring circuit (II)

13. 2.1. Squaring circuits

14. 2.1. Squaring circuits Current-input circuits

15. Squaring circuit (III)

16. 2.1. Squaring circuits

17. 2.1. Squaring circuits Current-input circuits Squaring circuit (IV)

18. 2.1. Squaring circuits

19. 2. Fundamental CMOS computational structures 2.2. Multiplier/divider circuits

20. 2.2. Multiplier/divider circuits Voltage-input circuits

21. Multiplier circuit (I)

22. Multiplier circuit (I) – block diagram 2.2. Multiplier/divider circuits

23. Multiplier circuit (I) – equivalent schematic 2.2. Multiplier/divider circuits

24. Realization of DA blocks 2.2. Multiplier/divider circuits

25. 2.2. Multiplier/divider circuits Voltage-input circuits Multiplier circuit (II)

26. Multifunctional core Multiplier circuit (II) 2.2. Multiplier/divider circuits

27. Multiplier schematic Multiplier circuit (II) 2.2. Multiplier/divider circuits

28. 2.2. Multiplier/divider circuits Current-input circuits

29. 2.2. Multiplier/divider circuits Current-input circuits Multiplier/divider circuit (III)

30. Multiplier/divider circuit (III) 2.2. Multiplier/divider circuits

31. 2.2. Multiplier/divider circuits Current-input circuits Multiplier/divider circuit (IV)

32. Multiplier/divider circuit (IV) 2.2. Multiplier/divider circuits

33. 2. Fundamental CMOS computational structures 2.3. Euclidean distance circuits

34. 2.3. Euclidean distance circuits Block diagram of the Euclidean distance circuit

35. Block diagram Block diagram of the Euclidean distance circuit 2.3. Euclidean distance circuits

36. Euclidean distance circuit

37. 2.3. Euclidean distance circuits

38. So: Similar: But: The I Y current can be expressed as follows: resulting: Euclidean distance circuit 2.3. Euclidean distance circuits

39. 2. Fundamental CMOS computational structures 2.4. Active resistor structures

40. 2.4. Active resistor structures Active resistor structure with positive equivalent resistance (I)

41. Active resistor structure with positive equivalent resistance (I) 2.4. Active resistor structures

42. 2.4. Active resistor structures Active resistor structure with negative equivalent resistance (I)

43. Active resistor structure with negative equivalent resistance (I) 2.4. Active resistor structures

44. 2.4. Active resistor structures Active resistor structure with positive equivalent resistance (II) - - block diagram

45. Resulting : Active resistor structure with positive equivalent resistance (II) - - block diagram 2.4. Active resistor structures

46. 2.4. Active resistor structures Active resistor structure with positive equivalent resistance (III) - - block diagram

47. Active resistor structure with positive equivalent resistance (III) - - block diagram 2.4. Active resistor structures

48. 3. Multifunctional structures

49. 3. Multifunctional structures Voltage-input circuits

50. 3. Multifunctional structures Voltage-input circuits Multifunctional circuit (I)

51. Multifunctional circuit (I) Nonlinear multifunctional core (NMC) 3. Multifunctional structures

52. Linear differential amplifier - block diagram 3. Multifunctional structures Multifunctional circuit (I)

53. 3. Multifunctional structures Multifunctional circuit (I) Realization of a linear differential amplifier - first implementation

54. Realization of a linear differential amplifier - second implementation 3. Multifunctional structures Multifunctional circuit (I)

55. Active resistor structure with positive equivalent resistance - - block diagram 3. Multifunctional structures Multifunctional circuit (I)

56. 3. Multifunctional structures Multifunctional circuit (I) Realization of an active resistor structure with positive equivalent resistance - first implementation

57. 3. Multifunctional structures Multifunctional circuit (I) Realization of an active resistor structure with positive equivalent resistance - second implementation

58. Active resistor structure with negative equivalent resistance - - block diagram 3. Multifunctional structures Multifunctional circuit (I)

59. 3. Multifunctional structures Multifunctional circuit (I) Realization of an active resistor structure with negative equivalent resistance - first implementation

60. Realization of an active resistor structure with negative equivalent resistance - second implementation 3. Multifunctional structures Multifunctional circuit (I)

61. Squaring circuit - block diagram 3. Multifunctional structures Multifunctional circuit (I)

62. 3. Multifunctional structures Multifunctional circuit (I) Realization of a squaring circuit – first implementation

63. Realization of a squaring circuit – second implementation 3. Multifunctional structures Multifunctional circuit (I)

64. Multiplier circuit (I) - block diagram 3. Multifunctional structures Multifunctional circuit (I)

65. 3. Multifunctional structures Multifunctional circuit (I) Realization of the multiplier circuit (I) - first implementation

66. 3. Multifunctional structures Multifunctional circuit (I) Realization of the multiplier circuit (I) - - second implementation

67. Multiplier circuit (II) - block diagram 3. Multifunctional structures Multifunctional circuit (I)

68. 3. Multifunctional structures Voltage-input circuits Multifunctional circuit (II)

69. Nonlinear multifunctional core (NMC) 3. Multifunctional structures Multifunctional circuit (II)

70. Realization of the nonlinear multifunctional core 3. Multifunctional structures Multifunctional circuit (II)

71. 3. Multifunctional structures Multifunctional circuit (II) Linear differential amplifier - block diagram

72. Realization of a linear differential amplifier 3. Multifunctional structures Multifunctional circuit (II)

73. 3. Multifunctional structures Multifunctional circuit (II) Squaring circuit - block diagram

74. Realization of the DA block 3. Multifunctional structures Multifunctional circuit (II) Multiplier circuit - block diagram

75. It results: 3. Multifunctional structures Multifunctional circuit (II) Multiplier circuit - circuit analysis

76. Realization of the multiplier circuit 3. Multifunctional structures Multifunctional circuit (II)

77. 3. Multifunctional structures Voltage-input circuits Multifunctional circuit (III)

78. Nonlinear multifunctional core (NMC) 3. Multifunctional structures Multifunctional circuit (III)

79. Realization of the linear differential amplifier 3. Multifunctional structures Multifunctional circuit (III)

80. 3. Multifunctional structures Multifunctional circuit (III) Realization of an active resistor with positive equivalent resistance

81. Realization of an active resistor with negative equivalent resistance 3. Multifunctional structures Multifunctional circuit (III)

82. Realization of the multiplier circuit It results: 3. Multifunctional structures Multifunctional circuit (III)

83. 3. Multifunctional structures Current-input circuits

84. 3. Multifunctional structures Current-input circuits Multifunctional circuit (IV)

85. Current-mode nonlinear multifunctional core (NMC) 3. Multifunctional structures Multifunctional circuit (IV)

86. Block diagram of the squaring circuit 3. Multifunctional structures Multifunctional circuit(IV)

87. Block diagram of the multiplier/divider circuit 3. Multifunctional structures Multifunctional circuit (IV)

88. Realization of the multiplier/divider circuit 3. Multifunctional structures Multifunctional circuit (IV)

89. 4. CMOS function synthesizers

90. 4.1. General function synthesizers

91. 4.1. General function synthesizers Third-order function synthesizer

92. General form of the approximation function: resulting: Taylor series of f(x) function: Third-order approximation function Third-order function synthesizer 4.1. General function synthesizers

93. Functional core (FC) of the function synthesizer Third-order function synthesizer 4.1. General function synthesizers

94. Block diagram of the function synthesizer 4.1. General function synthesizers Third-order function synthesizer

95. Circuit of the function synthesizer 4.1. General function synthesizers Third-order function synthesizer

96. 4.1. General function synthesizers Fourth-order function synthesizer

97. Funcţia de aproximare de ordin IV Fourth-order function synthesizer 4.1. General function synthesizers General form of the approximation function: Taylor series of f(x) function:

98. 4. CMOS function synthesizers 4.2. Exponential function synthesizers

99. 4.2. Exponential function synthesizers Approximation of the exponential function

100. Classical approximations of the exponential function: - Limited Taylor series: - Second-order approximation functions. Example: 4.2. Exponential function synthesizers Approximation of the exponential function

101. Proposed superior-order approximation functions: - Third-order approximations: - Fourth-order approximations: 4.2. Exponential function synthesizers

102. Exponential circuits using third-order approximation functions (a)

103. It results: 4.2. Exponential function synthesizers So, I OUT will be proportional (in a third-order approximation) with the exponential function:

104. 4.2. Exponential function synthesizers Exponential circuits using third-order approximation functions (b)

105. It results: 4.2. Exponential function synthesizers So, I OUT will be proportional (in a third-order approximation) with the exponential function:

106. 4.2. Exponential function synthesizers Exponential circuits using fourth-order approximation functions (a)

107. It results: 4.2. Exponential function synthesizers So, I OUT will be proportional (in a fourth-order approximation) with the exponential function:

108. 4.2. Exponential function synthesizers Exponential circuits using fourth-order approximation functions (b)

109. 4.2. Exponential function synthesizers

110. It results: So, I OUT will be proportional (in a fourth-order approximation) with the exponential function: So: 4.2. Exponential function synthesizers Exponential circuits using fourth-order approximation functions (b)

111. 4. CMOS function synthesizers 4.3. Gaussian function synthesizers

112. Approximation of the Gaussian function using Taylor series

113. Fundamental sixth-order Taylor series approximation function: 4.3. Gaussian function synthesizers Improved sixth-order Taylor series approximation function:

114. 4.3. Gaussian function synthesizers Block diagram of the Gaussian function circuit (I)

115. 4.3. Gaussian function synthesizers

116. Analysis of the Gaussian function circuit (I)

117. It results: 4.3. Gaussian function synthesizers

118. Block diagram of the Gaussian function circuit (II)

119. 4.3. Gaussian function synthesizers

120. Analysis of the Gaussian function circuit (II)

121. 4.3. Gaussian function synthesizers

122. It results: 4.3. Gaussian function synthesizers Analysis of the Gaussian function circuit (II)

123. 4.3. Gaussian function synthesizers Squaring circuit realization

124. It results: 4.3. Gaussian function synthesizers

125. 4. CMOS function synthesizers 4.4. Sinh (x) and tanh (x) function synthesizers

126. 4.4. Sinh (x) and tanh (x) function synthesizers Sinh (x) function synthesizer

127. Fifth-order approximation of the sinh(x) function Sinh (x) function synthesizer 4.4. Sinh (x) and tanh (x) function synthesizers Graphical representation of f(x) = sinh (x) and g 2 (x) functions Graphical representation of the approximation error Fifth-order approximation function:

128. Block diagram of the sinh (x) function synthesizer 4.4. Sinh (x) and tanh (x) function synthesizers Sinh (x) function synthesizer

129. Circuit of the sinh (x) function synthesizer 4.4. Sinh (x) and tanh (x) function synthesizers Sinh (x) function synthesizer

130. 4.4. Sinh (x) and tanh (x) function synthesizers Tanh (x) function synthesizer

131. Graphical representation of f(x) = tanh (x) and g 6 (x) functions Graphical representation of the approximation error Fifth-order approximation function: Fifth-order approximation of the tanh (x) function 4.4. Sinh (x) and tanh (x) function synthesizers Tanh (x) function synthesizer

132. Block diagram of the tanh (x) function synthesizer 4.4. Sinh (x) and tanh (x) function synthesizers Tanh (x) function synthesizer

133. 5. Conclusions

134. Advantages of analog computation: - Low-power operation - Real-time operation - Increased accuracy - Reduced complexity Optimized computational structures: - Fundamental CMOS computational structures - Multifunctional structures - CMOS function synthesizers

135. Thank you!