CIRF
Circuit Intégré Radio Fréquence
Lecture I
•Introduction
•Baseband Pulse Transmission
•Digital Bandpass Transmission
•Circuit Non-idealities Effect
Hassan Aboushady
University of Paris VI
CIRF
Circuit Intégré Radio Fréquence
Lecture I
•Introduction
•Baseband Pulse Transmission
•Digital Bandpass Transmission
•Circuit Non-idealities Effect
Hassan Aboushady
University of Paris VI
Disciplines required in RF design
H. Aboushady
University of Paris VI
References
• S. Haykin, “Communication Systems”, Wiley 1994.
• B. Razavi, “RF Microelectronics”, Prentice Hall, 1997.
• M. Perrott, “High Speed Communication Circuits and
Systems”, M.I.T.OpenCourseWare, http://ocw.mit.edu/,
Massachusetts Institute of Technology, 2003.
• D. Yee, “ A Design methodology for highly-integrated
low-power receivers for wireless communications”,
http://bwrc.eecs.berkeley.edu/, Ph.D. University of
California at berkeley, 2001.
H. Aboushady
University of Paris VI
CIRF
Circuit Intégré Radio Fréquence
Lecture I
•Introduction
•Baseband Pulse Transmission
•Digital Bandpass Transmission
•Circuit Non-idealities Effect
Hassan Aboushady
University of Paris VI
Matched Filter
Linear Receiver Model
• g(t) : transmitted pulse signal, binary symbol ‘1’ or ‘0’.
• w(t) : channel noise, sample function of a white noise process
of zero mean and power spectral density N0/2.
x(t ) = g (t ) + w(t ) , 0 ≤ t ≤ T
h(t )
y (t ) = g 0 (t ) + n (t )
• Filter Requirements, h(t) :
• Make the instantaneous power in the output signal g0(t) ,
measured at time t=T, as large as possible compared with
the average power of the output noise, n(t).
H. Aboushady
University of Paris VI
Maximize Signal-to-Noise Ratio
Linear Receiver Model
SNR =
instantaneous power in the output signal
average output noise power
SNR =
H. Aboushady
g 0 (T )
2
n 2 (t )
University of Paris VI
Math Review
Fourier Transform
∞
G( f ) =
∫
g (t ) e
− j 2π f t
dt
−∞
Inverse Fourier Transform
∞
g (t ) =
∫
G( f ) e
j 2π f t
df
−∞
Power Spectral Density of a Random Process SY(f)
applied to a Linear System H(f)
2
SY ( f ) = H ( f ) S X ( f )
H. Aboushady
University of Paris VI
Compute Signal-to-Noise Ratio
∞
g 0 (t ) =
∫
H ( f ) G( f ) e
j 2π f t
∫
2
g 0 (T ) =
df
−∞
2
N0
SN ( f ) =
H( f )
2
2
∞
H ( f ) G( f ) e
j 2π f T
df
−∞
∞
n 2 (t ) =
∫
N
n 2 (t ) = 0
2
S N ( f ) df
−∞
SNR =
∫
2
H ( f ) df
−∞
2
∞
∫
∞
H ( f ) G( f ) e
j 2π f T
df
−∞
N0
2
∞
∫
2
H ( f ) df
−∞
• Optimization Problem:
• For a given G(f) , find H(f) in order to maximize SNR.
H. Aboushady
University of Paris VI
Schwarz’s Inequality
• If we have 2 complex functions φ1(x) and φ2(x) in the real
variable x, satisfying the conditions:
∞
∞
∫
∫
2
φ1 ( x) dx < ∞
2
φ2 ( x) dx < ∞
−∞
−∞
then we may write that:
2
∞
∫
φ1 ( x) φ2 ( x) dx ≤
−∞
∞
∫
−∞
setting:
φ1 ( x) dx ∫ φ2 ( x) dx
2
2
φ1 ( x) = H ( f )
and
2
∫ H ( f ) G( f ) e
H. Aboushady
j 2π f T
df
φ2 ( x ) = G ( f ) e
∞
≤
∫
−∞
φ1 ( x) = k φ2* ( x)
where k : arbitrary constant
−∞
∞
−∞
iff
∞
j 2π f T
∞
2
H ( f ) df
∫
2
G ( f ) df
−∞
University of Paris VI
Matched Filter
2
SNR ≤
N0
∞
∫
2
SNRmax
G ( f ) df
−∞
∞
∫
G* ( f ) e
− j 2π f (T −t )
∞
∫
2
G ( f ) df
−∞
H opt ( f ) = k G * ( f ) e − j 2π
φ1 ( x) = k φ2* ( x)
hopt (t ) = k
2
=
N0
f T
∞
df
−∞
• for a real signal g(t) we have
G*(f)=G(-f)
hopt (t ) = k
∫
G (− f ) e
− j 2π f ( T −t )
df
−∞
hopt (t ) = k g (T − t )
• The impulse response of the optimum filter, except for the
scaling factor k, is a time-reversed and delayed version of the
input signal g(t)
H. Aboushady
University of Paris VI
Properties of Matched Filters
G0 ( f ) = H opt ( f ) G ( f )
hopt (t ) = k g (T − t )
H opt ( f ) = k G * ( f ) e − j 2π
= k G * ( f ) G ( f ) e − j 2π
f T
2
= k G ( f ) e − j 2π
f T
f T
• Taking the inverse Fourier transform at t=T:
∞
g 0 (T ) =
∫ G (f)e
∞
j 2π f T
0
df = k
−∞
N0
2
n (t ) =
2
−∞
∞
∫
∞
2
H ( f ) df
−∞
SNRmax
H. Aboushady
∫
Where E is the
G ( f ) df = k E energy of the
pulse signal g(t)
2
n 2 (t ) =
2
N0 2
N
k
G ( f ) df = k 2 0 E
2 −∞
2
∫
k 2E2
2E
=
=
N
k 2 0 E N0
2
University of Paris VI
Matched Filter for Rectangular Pulse
hopt (t ) = k g (T − t )
h(t) for a rectangular Pulse:
g(-t)
g(t)
T
t
T
t
h(t) = g(T-t)
T
t
Filter Output g(t)*h(t):
g(t)
Implementation:
H. Aboushady
University of Paris VI
Error Rate due to Noise
x(t)
In the interval
0 ≤ t ≤ Tb , the received signal:
⎧ + A + w(t ) , symbol '1' was sent
x(t ) = ⎨
⎩− A + w(t ) , symbol '0' was sent
Tb is the bit duration, A is the transmitted pulse amplitude
• The receiver has prior knowledge of the pulse shape but not
its polarity.
• There are two possible kinds of error to be considered:
(1) Symbol ‘1’ is chosen when a ‘0’ was transmitted.
(2) Symbol ‘0’ is chosen when a ‘1’ was transmitted.
H. Aboushady
University of Paris VI
Error Rate due to Noise
x (t ) = − A + w(t ) , 0 ≤ t ≤ Tb
Suppose that symbol ‘0’ was sent:
Tb
Y=
The matched filter output is:
1
x(t ) dt = − A +
Tb
∫
0
Tb
∫ w(t ) dt
0
Y is a random variable with Gaussian distribution and a mean of -A.
1
The variance of Y: σ = (Y + A) = 2
Tb
2
Y
2
∫∫
Tb
Tb
0
0
RW (t , u ) dt du
Where RW(t,u) is the autocorrelation function of the white noise w(t) .
Since w(t) is white with a PSD of N0/2 :
RW (t , u ) =
N0
δ (t − u )
2
N0
σ =
2 Tb
2
Y
H. Aboushady
University of Paris VI
PDF: Probability Density Function
• Gaussian Distribution:
fY ( y ) =
1
σY
Normalized PDF
µY= 0 and σY=1
⎡ ( y − µY ) 2 ⎤
exp ⎢−
⎥
2
2
σ
2π
Y
⎦
⎣
N0
• Symbol ‘0’ was sent: µY = − A , σ =
Tb
Pe 0 = P ( y > λ symbol '0' was sent)
2
Y
∞
=
∫
fY ( y 0) dy
λ
2
• Symbol ‘1’ was sent: µY = + A , σ Y =
Pe1 = P ( y < λ symbol '1' was sent)
N0
Tb
λ
=
H. Aboushady
∫
−∞
fY ( y 1) dy
University of Paris VI
BER in a PCM receiver
Pe 0 =
1
π N 0 / Tb
∫
∞
λ
⎡ ( y + A) 2 ⎤
exp ⎢−
⎥ dy
⎣ N 0 / Tb ⎦
• let λ=0 and the probabilities of binary symbols: p0 = p1 = 1/2 .
z=
y+A
N 0 / Tb
Pe 0 =
1
π
∫
[ ]
∞
exp − z 2 dz
Eb / N 0
• where Eb =A2Tb , is the transmitted signal energy per bit.
• the complementary error function:
⎛ Eb ⎞
1
⎟
Pe1 = Pe 0 = erfc⎜
⎜ N0 ⎟
2
⎝
⎠
H. Aboushady
erfc(u ) =
1
π ∫
∞
u
[ ]
exp − z 2 dz
University of Paris VI
BER in a PCM receiver
Pe = p0 Pe 0 + p1Pe1
Pe 0 = Pe1
1
p0 = p1 =
2
Pe = Pe 0 + Pe1
⎛ Eb ⎞
1
⎟
Pe = erfc⎜
⎜ N0 ⎟
2
⎝
⎠
H. Aboushady
University of Paris VI
CIRF
Circuit Intégré Radio Fréquence
Lecture I
•Introduction
•Baseband Pulse Transmission
•Digital Bandpass Transmission
•Circuit Non-idealities Effect
Hassan Aboushady
University of Paris VI
Why Modulation?
• In wired systems, coaxial lines exhibit superior shielding
at higher frequencies
• In wireless systems, the antenna size should be a
significant fraction of the wavelength to achieve a
reasonable gain.
• Communication must occur in a certain part of the
spectrum because of FCC regulations.
• Modulation allows simpler detection at the receive end in
the presence of non-idealities in the communication
channel.
H. Aboushady
University of Paris VI
Message Source
• mi : one symbol every T seconds
• Symbols belong to an alphabet of M symbols: m1, m2, …, mM
• Message output probability:
P ( m1 ) = P ( m2 ) = ... = P (mM )
pi
= P (mi ) =
1
M
• Example: Quaternary PCM, 4 symbols: 00, 01, 10, 11
H. Aboushady
University of Paris VI
Transmitter
• Signal Transmission Encoder: produces a vector si made up of N
real elements, where N ≤ M .
• Modulator: constructs a distinct signal si(t) representing mi of
duration T .
T
• Energy of si(t) :
Ei =
∫
si2 (t ) dt , i = 1, 2, ..., M
0
• si(t) is real valued and transmitted every T seconds.
H. Aboushady
University of Paris VI
Examples of Transmitted signals: si(t)
• The modulator performs a step change in the amplitude, phase
or frequency of the sinusoidal carrier
• ASK:
Amplitude Shift Keying
• PSK:
Phase Shift Keying
• FSK:
Frequency Shift Keying
Special case: Symbol Duration T = Bit Duration, Tb
H. Aboushady
University of Paris VI
Communication Channel
• Two Assumptions:
•The channel is linear
≤ (no distortion).
• si(t) is perturbed by an Additive, zero-mean, stationnary,
White, Gaussian Noise process (AWGN).
• Received signal x(t) :
⎧ 0≤t ≤T
x(t ) = si (t ) + w(t ), ⎨
⎩i = 1, 2,..., M
H. Aboushady
University of Paris VI
Receiver
•TASK: observe received signal, x(t), for a duration T and make a
best estimate of transmitted symbol, mi .
•Detector: produces observation vector x .
•Signal Transmission Decoder: estimates m̂ using x, the
modulation format and P(mi) .
• The requirement is to design a receiver so as to minimize
the average probablilty of symbol error:
M
Pe =
∑ P(mˆ = m ) P(m )
i
i
i =1
H. Aboushady
University of Paris VI
Coherent and Non-Coherent Detection
• Coherent Detection:
- The receiver is time synchronized with the transmitter.
- The receiver knows the instants of time when the
modulator changes state.
- The receiver is phase-locked to the transmitter.
• Non-Coherent Detection:
- No phase synchronism between transmitter and receiver.
H. Aboushady
University of Paris VI
Gram-Schmidt Orthogonalization Procedure
• we represent the given set of realvalued energy signals s1(t), s2(t), … ,
sM(t), each of duration T:
⎧ 0≤t ≤T
si (t ) =
sij φ j (t ) , ⎨
j =1
⎩i = 1, 2,..., M
N
∑
• where the coefficients of the
expansion are defined by:
Mod
u
lato
r
⎧i = 1, 2,..., M
sij = si (t ) φ j (t ) dt , ⎨
⎩ j = 1, 2,..., N
0
T
∫
• the real-valued basis functions φ1(t),
φ2(t), … , φN(t) are orthonormal:
⎧1
φi (t ) φ j (t ) dt = ⎨
⎩0
0
T
∫
H. Aboushady
if
i= j
if
i≠ j
D et
ecto
r
University of Paris VI
Coherent Detection of Signals in Noise
• Signal Vector si :
⎡ si1 ⎤
⎢ ⎥
si 2 ⎥
⎢
si =
, i = 1,2,..., M
⎢M ⎥
⎢ ⎥
⎣⎢ siN ⎦⎥
• Observation vector x :
x = si + w , i = 1, 2,..., M
⎧ 0≤t ≤T
x(t ) = si (t ) + w(t ), ⎨
⎩i = 1, 2,..., M
• w(t) is a sample function of
an AWGN with power spectral
density N0 /2.
where w is the noise vector.
H. Aboushady
University of Paris VI
Coherent Binary PSK:
• M=2, N=1
s1 (t ) =
0 ≤ t ≤ Tb
2 Eb
cos( 2πf c t )
Tb
2 Eb
cos( 2πf ct )
s2 (t ) = −
Tb
• To ensure that each transmitted bit contains an integral number of
cycles of the carrier wave, fc =nc/Tb, for some fixed integer nc.
2
• One basis function: φ1 (t ) =
cos( 2πf c t ) , 0 ≤ t ≤ Tb
Tb
• Signal constellation consists of
two message points:
Tb
∫
s11 = s1 (t ) φ1 (t ) dt = Eb
0
Tb
∫
s21 = s2 (t ) φ1 (t ) dt = − Eb
0
H. Aboushady
University of Paris VI
Generation and Detection of Coherent Binary PSK
• Assuming white Gaussian Noise with
PSD = N0/2 ,
The Bit Error Rate for coherent binary PSK is:
⎛ Eb ⎞
1
⎟
Pe = erfc⎜
⎜ N0 ⎟
2
⎝
⎠
Binary PSK Transmitter
Coherent Binary PSK Receiver
H. Aboushady
University of Paris VI
Coherent QPSK:
• M=4, N=2:
⎧ 2E
π⎤
⎡
⎪
si (t ) = ⎨ T cos ⎢⎣ 2πf ct + (2i − 1) 4 ⎥⎦ , 0 ≤ t ≤ T
⎪0
, elsewhere
⎩
2E
π
cos( 2π f ct + )
s1 (t ) =
T
4
2E
π
s2 (t ) =
cos( 2π f ct + 3 )
T
4
2E
π
s3 (t ) =
cos( 2π f ct + 5 )
T
4
2E
π
s4 (t ) =
cos( 2π f ct + 7 )
T
4
2
cos( 2π f ct ) , 0 ≤ t ≤ T
T
2
φ2 (t ) =
sin( 2π f ct ) , 0 ≤ t ≤ T
T
University of Paris VI
• Two basis function: φ1 (t ) =
H. Aboushady
Constellation Diagram of Coherent QPSK System
⎛ Eb ⎞
1
⎟
Pe = erfc⎜
⎜ N0 ⎟
2
⎝
⎠
H. Aboushady
University of Paris VI
QPSK waveform: 01101000
H. Aboushady
University of Paris VI
Generation and Detection of Coherent QPSK Signals
QPSK Transmitter
Coherent QPSK Receiver
H. Aboushady
University of Paris VI
Power Spectra of BPSK ,QPSK and M-ary PSK
• Symbol Duration:
T = Tb log 2 M
• Power Spectral Density of
an M-ary PSK signal:
S B ( f ) = 2 E sinc 2 (Tf )
8-PSK
= 2 Eb log 2 M sinc 2 (Tb f log 2 M )
S B ( f ) = 6 Eb sinc 2 (3Tb f )
QPSK
S B ( f ) = 4 Eb sinc 2 (2Tb f )
BPSK
H. Aboushady
S B ( f ) = 2 Eb sinc2 (Tb f )
University of Paris VI
CIRF
Circuit Intégré Radio Fréquence
Lecture I
•Introduction
•Baseband Pulse Transmission
•Digital Bandpass Transmission
•Circuit Non-idealities Effect
Hassan Aboushady
University of Paris VI
QPSK Receiver
φ1 (t ) =
2
cos(2π f c t )
T
Threshold = 0
T
T
Decision
Device
∫ dt
path I
0
011011
Binary Data
Multiplexer
antenna
path Q
T
Decision
Device
∫ dt
0
T
φ1 (t ) =
2
sin( 2π f c t )
T
Threshold = 0
01
Q
11
I
QPSK Constellation Diagram
00
H. Aboushady
10
University of Paris VI
Receiver Circuit Non-Idealities
• Circuit Noise (Thermal, 1/f)
• Gain Mismatch
• Phase Mismatch
• DC Offset
• Frequency Offset
• Local Oscillator phase noise
H. Aboushady
University of Paris VI
Circuit Noise
Q
Circuit Noise:
• Thermal Noise
• Resistors
• Transistors
I
• Flicker (1/f) Noise
• MOS transistors
H. Aboushady
University of Paris VI
Gain Mismatch
A(1+α/2)
A(1-α/2)
2
Q
1.5
1
0.5
I
0
-0.5
-1
-1.5
-2
-2
H. Aboushady
-1.5
-1
-0.5
0
0.5
1
1.5
2
University of Paris VI
Phase Mismatch
2
Q
1.5
1
0.5
I
0
-0.5
-1
-1.5
-2
-2
H. Aboushady
-1.5
-1
-0.5
0
0.5
1
1.5
2
University of Paris VI
DC Offset
offset
I
Q
offset
2
Q
1.5
1
0.5
I
0
-0.5
-1
-1.5
-2
-2
H. Aboushady
-1.5
-1
-0.5
0
0.5
1
1.5
2
University of Paris VI
Frequency Offset
2
Q
1.5
1
0.5
I
0
-0.5
-1
-1.5
-2
-2
H. Aboushady
-1.5
-1
-0.5
0
0.5
1
1.5
2
University of Paris VI
Local Oscillator Phase Noise
H. Aboushady
University of Paris VI
Reciprocal Mixing
H. Aboushady
University of Paris VI