f RF f LO f RF ±f LO Ιδανικός μείκτης RF Είσοδος f RF f RF ± f LO IF Έξοδος f LO LO Είσοδος f RF f LO

(ω RF t) (ω LO t) = 1 2 [(ω RF + ω LO )t + (ω RF ω LO )t] RF LO IF f RF ± f LO 0 180

+1 RF IF 1 LO LO IF R out RF

IF Q3 Q4 Q5 Q6 LO Q1 Q2 RF I

cos(ωk) + 1 1 x k T S ZOH x(t) u k f S = 1/T S

{ k } k = [(Ωk) k ] ±1 { k } G : [ 1, 1] [0, 1] [ 1, 1] k Z u [ 1, 1] ( k u) = G(u) { k } [ 1, 1] (Ωk) Ω/(2π) Ω = 2πw/q w 0 < w < q/2 [ ( k = 2π kw q ) k ] { k } (2πkw/q) q/ (q, w) { k } q q/ (q, w) { k } q {x k } r x (n, m) = E{ n m } { k } s x (ω) r x (k) = 1 q q 1 m=0 r x (k + m, m) s x (ω) = k= r x (k)e ikω. r x (k) G : [ 1, 1] [0, 1] G(u) = 1 2 + 1 2 a j T j (u). j=0

a j a 0 = 2 π 1 1 G(u) 1 u 2 u 1 a j>0 = 4 π 1 1 G(u)T j (u) u 1 u 2 G G [ 1, 1] G a j G G G( 1) = 0, G(1) = 1 G (u) 0 u [ 1, 1] T j (±1) = (±1) j j = 0, 1, 2,... T j (u) = ju j 1(u) j = 1, 2, 3,... U j j ( 1) j a j = 1, j=0 a j = 1, j=0 ja j U j 1 (u) 0 u [ 1, 1] j=1 r x (k) a j j = 0, 1, 2,... r x (k) { k } δ 0 = 1 δ k 0 = 0 r x (k) = a 2 a 2 ( ) j 2πkjw 0 + 2 + 1 a 2 0 q j=1 j=1 a 2 j 2 δ k r x a 2 0 (2πkw/q) k = 0 j a j G { k } (2πkw/q) ω [0, 2π] (2πkjw/q) = (2πk ((jw) q) /q)

x(t) x(t) = k= ( ) t k p k T S T S = 1/f S p(t) t [0, 1) x(t) S x (f) = T S 2 (ft S ) s x (2πfT S ) s x r x T S 2 (ft S ) p(t) (w, q) = 1 x(t) ( ) ) f S x (f) = (S 2 HA (f) + S N (f) + S DC (f) f S S HA S N S DC S HA (f) = 1 4 h=1 S N (f) = 1 1 a 2 0 1 f S 2 ( b h [δ f h ) ( q f S + δ f + h )] q f S j=1 a 2 j S DC (f) = b 0 + 3a 2 0 4 δ(f) h = 0, 1, 2,... ±(h/q) f S b h /4 b h = r= a 2 I(h,r) I(h, r) = j 1 h + qr j 1 k 1 wj 1 +qk 1 = 1 b w ±(w/q) f S b w = r= a 2 1+qr

a j ±(h/q) f S a 2 I(h,r) 0 a j b h (j 1, k 1 ) wj 1 +qk 1 = 1 (w, q) = 1 (j 1, k 1 ) j 1, k 1 j 1 j 1 h + qr (w/q) f S 2 (f/f S ) b w /4 2 (f/f S ) S N (f) DR = 10 10 2 ( f f S ) bw4 2 ( f f S ) S N (f) b w S N (f) r= DR = 10 10 1 a 2 0 1 2 a 2 1+qr a 2 j j=1 + 10 10 f S. b h h = 0, 1, 2,... 10 10 f S f S

w = 25 q = 64 G(u) u [ 1, 0) u (0, 1] k = 0 G(u) 1 0 u G G u = 0 u 0 a 2k = 0 a 2k+1 = 4( 1) k /((2k + 1)π) k = 0, 1, 2,... 2 (f/f S ) Normalized Power (db) 0-10 -20-30 -40-50 f S = 1 GHz RBW = 1000 Hz -60 0 0.2 0.4 0.6 0.8 1 f/f S w = 25 q = 64 2 (f/f S ) w = 25 q = 64 { k } G (u) = 1/2 [ 1, 1] G(u) = (u + 1)/2 T 1 (u) = u a 0 = 0 a 1 = 1 a k = 0 k = 2, 3, 4,...

G(u) G'(u) 1 1 2 1 0 1 u (α) 1 0 1 u 1 0 1 t (β) b h 25j 1 + 64k 1 = 1 (j 1, k 1 ) = ( 23, 9) b h h = 0, 1, 2,... r Z I(h, r) = j 1 h + 64r = 1 a 1 = 1 a 0 = 0 a k = 0 k = 2, 3, 4,... a 1 b h 25j 1 +64k 1 = 1 j 1 h+64r = 1 (h, r) = (25, k 1 ) (h, r) = ±(25, 9) + r(64, 23), r Z h 0 b h b 25 b 64n±25 n = 1, 2, 3,... b 25 = b 64n±25 = 1 n = 1, 2, 3,... S HA (f) = 1 4 h=25, 64n±25 n=1,2,3,... [ ( δ f h ) ( q f S + δ f + h )] q f S S N (f) = 1/(2f S ) S DC (f) = 0 2 (f/f S ) [0, f S ] (25/64) f S ((64 25)/64) f S

0 f S = 1 GHz Normalized Power (db) -10-20 -30-40 -50 RBW = 3125 Hz N av = 10 52 db -60 0 0.2 0.4 0.6 0.8 1 f/f S w = 25 q = 64 2 (f/f S ) f S = N = 10 { k } w, q a 0 = 0 a 1 = 1 a k = 0 k = 2, 3, 4,... DR = 10 10 (f S ). f S = DR 10 10 (RBW ) RBW =

Switching Mixer RF Input Matching Network Matching Network IF Output Single-Bit Output Direct All-Digital Synthesizer

n f clk Register Random Number Generator n n w n 1 n+1 Truncation mod n to MSB Output n 1 [0, f clk ] F (u) = ( 1 + ( πu 2 n 1 π ) ) u = 0, 1, 2,..., (2 n 1 1) 2

f clk f LO = (w/2 n )f clk w n 1 0 < w < 2 n 1 [0, f clk /2] f clk /2 n v(t) ±1 ( ) [ f 1 ( S v (f) = 2 f clk 4 δ f ± w ) 2 n f clk + 1 ] 2f clk f [ f clk, f clk ] f clk 2 (f/f clk ) g(t) y(t) = g(t)x(t) x(t) x(t) = [v(t) + 1]/2 ±1 y(t) R y (t, t + τ) = 1 4 E { [v(t) + 1 ] g(t) [ v(t + τ) + 1 ] g(t + τ) } g(t)

R y (t, t + τ) = = = g(t)g(t + τ) { } E v(t)v(t + τ) + v(t) + v(t + τ) + 1 4 g(t)g(t + τ) { } E v(t)v(t + τ) + 1 4 g(t)g(t + τ) [ ] 1 + R v (t, t + τ) 4 E { v(t) } = 0 t y(t) R y R y (τ) = T [ 1 2T T T R y (t, t + τ) t ] g(t) = A (2πf g t) R y (τ) = A2 8 (2πf gt) [ 1 + R v (τ) ] y(t) R y (τ) (2πf g t) [ 1 + R v (τ) ] [δ(f) + S v (f)] S v S y (f) = A2 [ ] [ ] δ(f f g ) + δ(f + f g ) δ(f) + S v (f) 16 = A2 [ ] δ(f f g ) + δ(f + f g ) + S v (f f g ) + S v (f + f g ) 16 S y (f) = A2 [ ] δ(f f g ) + δ(f + f g ) + 16 ( f fg + A2 16 2 + A2 16 2 f clk ( f + fg f clk )[ 1 ( 4 δ f f g ± w ) 2 n f clk + 1 ] 2f clk )[ 1 ( 4 δ f + f g ± w ) 2 n f clk + 1 ] 2f clk

f = w 2 n f clk f g A2 64 2 ( w 2 n ) A 2 32f clk [ ( w 2 2 n 2f ) ( )] g w2 + 2 f n clk ( ) 2 w 2 2f g NP ower = 10 10 1 + n f ( ) clk 10 10 (f clk ) +. 2 w 2 n w/2 n f g /f clk NP ower 6 10 10 (f clk ) f clk = f LO = f RF =. f LO f RF f clk = f LO =. f RF = 0 f LO f RF

64 db f clk = f RF =. f LO = f clk = f RF = f LO =.

77 db f clk = f RF f LO f RF = f LO =.

RF Input s 2 C i s 1 s' 3 s 3 Output C i s 1 ΟΝ RF Input s 2 ΟΝ C i

RF Input s 2 C i s 1 s' 3 s 3 Output s 1 C i s 1 ΟΝ s 2 s' 3 RF Input C i s 1 s 3 Output RF Input s 2 ΟΝ C i C i s 1 ΟΝ s 2 C i C i RF Input s 2 ΟΝ C i s 3 ΟΝ Output (normal polarity) C i C i s 3 ΟΝ Output Output (normal polarity) s' s 3 ΟΝ 3 s 3 (inverted polarity) C i s' 3 ΟΝ Output (inverted polarity)

RF Input s 2 ΟΝ C i RF Input s 2 ΟΝ C i C i s 3 ΟΝ Output (normal polarity) C i s 3 ΟΝ Output (normal polarity) C i s' 3 ΟΝ Output (inverted polarity) C i s' 3 ΟΝ Output (inverted polarity) ±1 m m + 1 R 0 t t 1 v 1, v 2, v 3,..., v m m v out

v 1 (t) v 2 (t) v 3 (t) v m (t) v out (t) = 1 RC 1 0 0 0 0 v 1 (t) 0 1 0 0 0 v 2 (t) 0 0 1 0 0 v 3 (t) 0 0 0 1 0 v m (t) 0 0 0 0 0 v out (t) 0 t t 1 v 1 (t) v 2 (t) v 3 (t) = v m (t) v out (t) RC 0 0 0 0 v 1 (0) 0 e t RC 0 0 0 v 2 (0) 0 0 e t RC 0 0 v 3 (0) 0 0 0 e t RC 0 v m (0) 0 0 0 0 1 v out (0) e t t 1 < t t 2 u(t) f S u(t) u(t) v 1 (t) v 2 (t) v 3 (t) v m (t) v out (t) = 1 2RC 1 0 0 0 0 v 1 (t) 0 1 0 0 0 v 2 (t) 0 0 1 0 0 v 3 (t) 0 0 0 1 0 v m (t) 0 0 0 0 0 v out (t) + 1 2RC 1 1 1 u(t) 1 0 v(t) = Av(t) + bu(t)

(t 1, t 2 ] t v(t) = e A(t t1) v(t 1 ) + e A(t τ) bu(τ)dτ t 1 u(t) u(t 1 ) v 1 (t) v 2 (t) v 3 (t) v m (t) v out (t) = e t t 1 2RC 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 e t t 1 2RC v 1 (t 1 ) v 2 (t 1 ) v 3 (t 1 ) + 2RC(1 e t t 1 v m (t 1 ) v out (t 1 ) 2RC ) 1 1 1 u(t 1 ) 1 0 t 2 < t t 3 p i i +1 1 k C out C i C out p = [p 1 p 2... p m ] m = 2 b b ±1 v 1 (t) v 2 (t) v 3 (t) v m (t) v out (t) = 1 2RC 1 0 0 0 p 1 v 1 (t) 0 1 0 0 p 2 v 2 (t) 0 0 1 0 p 3 v 3 (t) 0 0 0 1 p m v m (t) kp 1 kp 2 kp 3 kp m mk v out (t)

A A = 1 2RC I ap p mk e At A n e At p p = m A 2 A 2 = ( 1 2RC ) 2 I + kpp (1 + mk)p (k + mk 2 )p mk + (mk) 2 A n A n = ( 1 2RC ) n I + α n pp γ n p β n p δ n A n+1 = A n A A n+1 = ( 1 2RC ) n+1 I + (α n kβ n )pp ( 1 mα n + mkβ n )p (γ n kδ n )p mγ n + mkδ n

α n+1 = 1 k α n + 0 β n+1 m mk β n 1 γ n+1 = 1 k γ n δ n+1 m mk δ n }{{} X γ n = X n 1 γ 1 = X n 1 k mk δ n δ 1 α n = X n 1 α n 2 1 + β n β 1 j=0 X j 0 1 n 2 X j = (X n 1 I)(X I) 1 j=0 α n = X n 1 α 1 + (X n 1 I)(X I) 1 0 1 β n β 1 X X n 1 = (1 + mk) n 2 X (X I) 1 = 1 1 mk k mk m 0

α n = 1 [ m (1 + mk) n 1 1 ] β n (1 + mk) n 1 γ n k(1 + mk)n 1 = mk(1 + mk) n 1 δ n A n A n = ( 1 2RC ) n I + 1 [ m (1 + mk) n 1 1 ] pp (1 + mk) n 1 p k(1 + mk) n 1 p mk(1 + mk) n 1 e At e At = I + At + 1 2! A2 t 2 + 1 3! A3 t 3 +... = n=0 1 n! An t n

1 quantized sinewave 0 1 dithering sequence to be added before quantization -1 t 0-1 t b = f S = RBW = f LO = 0.212f S f RF = 0.2f S R = C i = C out =

0 Normalized Power (db) -10-20 -30-40 -50-60 -70-80 68 db b = 2 RBW = 100 Hz f S = 1 GHz N av = 10-90 -100-110 -120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S 0-10 -20-30 b = 3 RBW = 100 Hz f S = 1 GHz N av = 10 Normalized Power (db) -40-50 -60-70 -80 77 db -90-100 -110-120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S 0-10 -20-30 b = 4 RBW = 100 Hz f S = 1 GHz N av = 10 Normalized Power (db) -40-50 -60-70 -80 84 db -90-100 -110-120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S [0, f S ] b = 2, 3, 4 f S = RBW = f LO = 0.212f S f RF = 0.2f S R = C i = C out =

0-10 -20-30 b = 5 RBW = 100 Hz f S = 1 GHz N av = 10 Normalized Power (db) -40-50 -60-70 -80 91 db -90-100 -110-120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S 0-10 -20-30 b = 6 RBW = 100 Hz f S = 1 GHz N av = 10 Normalized Power (db) -40-50 -60-70 -80 97 db -90-100 -110-120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S 0-10 -20-30 b = 7 RBW = 100 Hz f S = 1 GHz N av = 10 Normalized Power (db) -40-50 -60-70 -80 102 db -90-100 -110-120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f/f S [0, f S ] b = 5, 6, 7 f S = RBW = f LO = 0.212f S f RF = 0.2f S R = C i = C out =