DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/
OULINE Sampled data system Data reconstruction he z-ransform he relationship between F(z) and F(ω) Approximate mappings F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/
SAMPLED DAA SYSEMS A sampled data system processes signals in discrete time steps. Its input and its output are made of a sequence of numbers (digital sampled data system) or of a sequence of analog values (analog sampled data system). Such a type of signals are known as discrete time-signals. Antialiasing Filter IS Sampled Data Proc. Sample & Hold Œ Reconstr. Filter ª Œ ª F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/3
Is represents an ideal sampler with sampling period. It gives at its output the values of the input signal at the instants n. f * () t ft ()δt ( n) he following aspects are relevant: he sampling operation. he aliasing effect. he knowledge of the z-transform. F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/4
SAMPLING OF A CONINUOUS-IME SIGNAL f(0) f() f() f(t) 0 3 4 5 st () δt ( n) -- e jnω s t Where ω s π/ is the angular sampling rate F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/5
f * () t ft ()δt ( n) -- ft ()e jnω s t f(t) f(t)* f(t)s(t) if we take the Laplace transform of the two expressions of f*(t), we get: s(t) Σ δ(t-nt) t F * () t fn ( )e sn Poissons formula: F * ( s) -- Fs jnω ( s ) fn ( )e snt -- Fs jnω ( s ) F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/6
F(ω) F(ω) ω ω n n n n ω ω ω s ω s ω s ω s Shannon sampling theorem: A band limited signal x(t), with Fourier spectrum X(jω) that vanishes for ω > ω s / is described by its sample values x(n), where π/ω s. x(t) is given by: xt () xn ( ) sin( ω s ( t n) ) ---------------------------------------------- ω s ( t n) F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/7
DAA RECONSRUCION CARDINAL HOLD rt () sin( ω s t ) ---------------------------- ω s t x* τ rt ( τ) dτ x* τ δ( τ n) ------------------------------------------ sin( ω s ( t τ) ) d ω s ( t τ) τ x* ( n) sin( ω s ( t n) ) ---------------------------------------------- ω s ( t n) xt () F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/8
he simplest reconstruction filter is the hold. Its impulse response h(t) is given by: h(t) for 0 < t < 0 for t he frequency response of the hold is: 0 h(t) t H( ω) e jω ---------------------- e jω jω -- sin( ω ) ---------------------------- ω y(t) f*(t) h(t) f(t) t F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/9
Cardinalhold Sam ple-and-hold -4 π/ - π/ 0 π/ 4 π/ 3p p p Phase response -4 -p π / - π / 0 π / 4 π / - p -3 p Frequency response of the cardinal hold and of the hold F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/0
HE Z-RANSFORM Fz ( ) fn ( )z n Example: he z-transform of the function u(n) obtained by sampling the step function u(t) is: Uz ( ) un ( )z n z n It is know that the geometric series z -n converges for z >. However, U(z) is assumed as the z transform of u(n) even inside the unit circle in the z-plane. 0 ---------------- z F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/
Properties of the z-transform: he z-transform is linear: zc ( f ( n) + c f ( n) +...) c F ( z) + c F ( z) +... Convolution of two signals: z{ f ( n) f ( n)} F ( z) F ( z) Delayed signals: z{ f( n k)} Fz ( )z k F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/
Frequency translation: G*(ω) F*(ω ω 0 ) Initial value theorem: f0 ( ) lim Fz ( ) z Final value theorem: lim fn ( ) lim ( z )Fz ( ) n z F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/3
HE RELAIONSHIP BEWEEN F(Z) AND F(S) z e s he relation defines a mapping between two complex planes the s- plane and the z-plane. s σ + jω z e σ e jω he points in the left-half s-plane map inside the unity circle in the z- plane; the points in the right-half s-plane map outside the unity circle in the z-plane. he points of the imaginary axis jω of the s-plane map the unity circle. he mapping between the s and the z plane is not univocal: points in the s-plane with the same real part and imaginary part which differ by nπ (n integer) get mapped on the same point of the z-plane. F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/4
Im(s) jω Im(z) B B B j6π/ j4π/ jπ/ ω β ω σ Re(s) σ σ mod e Re(z) S - plane Z - plane A continuous time system is stable if the poles of its transfer function lie in the left half s-plane. A sampled data system is stable if the poles of its sampled data transfer function are inside the unity circle. he frequency response of a continuous time system is get by evaluating the pulse transfer function H(s) on the imaginary axis. he frequency response of a discrete time system is get by evaluating its pulse transfer function H(z) on the unity circle. F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/5
APPROXIMAE MAPPINGS s ( ) a m s m + a m s m +...+a 0 ----------------------------------------------------------------------- s n + b n s n +...+b 0 z ( ) α m z m + α m z m +...+α 0 ------------------------------------------------------------------------ z n + β n z n +...+β 0 Euler forward difference Euler backward difference z e s + s z ---------- e s --------------- s F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/6
Bilinear z e s ---------------- e s ----------------------- + s s Lossless Discrete (LD) z z e s e s ( + s ) ( s ) s df ---- dt n fn ( + ) fn ( ) ------------------------------------------- df ---- dt n fn ( ) fn ( ) ------------------------------------------- F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/7
aking the transform, we get: -- df df fn ( ) fn ( ) ---- + ---- dt dt ------------------------------------------- df ---- dt n n n fn ( + ) fn ( ) -------------------------------------------------------------------- sf( s) Fz ( ) z ----------- sf( s) Fz ( ) z ---------------- z ----------- -- --sf( s) ( + e s ) Fz ( ) z ---------------- sf( s) Fz ( ) z z ------------------------------ F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/8
Forward Euler s z ----------- Backward Euler s z ----------- z -- ( z ) Bilinear transformation s -- z ----------- z+ -- z ----------------- + z Lossless Discrete s z z ------------------------------ z ----------- z -- F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/9
jω e jω -- -------------------- e jω j + -- Ω tan------- Bilinear transf. jω e jω e jω -- ------------------------------------------ j -- Ω sin------- LD transf. 3 w Bilinear LD 0 p W Frequency warping of the bilinear and LD transformation F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/0
s - path b Forward Euler Im(s) jω s - path a Re(s) σ z - path b Im(z) z - path a Re(z) Im(s) jω s - path Re(s) σ Bilinear Im(z) z - path Re(z) S - plane Z - plane S - plane Z - plane Im(s) jω S - plane Backward Euler s - path b Re(s) σ Im(z) Z - plane z - path a Re(z) S - plane Im(s) jω s - path Loss less Discrete Re(s) σ Im(z) Z - plane z - path Re(z) F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete time signal processing 7/