Berkeley
Wireless Receiver Architectures
Prof. Ali M. Niknejad and Dr. Osama Shana’a
U.C. Berkeley
c 2014 by Ali M. Niknejad
Copyright Niknejad
Advanced IC’s for Comm
Outline
Review of key receiver specs
Complex baseband equivalent of a bandpass signal
Superheterodyne
Image Rejection Architectures
Direct-conversion
References
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Review of Receiver Specifications
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Receiver Specifications
Sensitivity, the weakest signal that can be detected by a
receiver. Set by amount of Amplificaiton and Noise
Selectivity, or the ability to select one frequency band in the
presence of interfering signals in nearby frequency bands.
Determined by the amount of Filtering and the receiver
Linearity
Dynamic Range, or the linear range of a receiver signal that
can be processed with minimal impact on the detection bit
error rate (BER), related to the Linearity and Noise in the
receiver. Greatly improved with Variable Gain Control (or
Programmable Gain Control)
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Receiver Goal #1: Amplification
LNA
ADC
A detector works well with a fairly strong signal. For instance, if the
input referred noise/offset/LSB is 10’s mV, the input signal should
be 10X or more larger.
The weakest signals can be -100 dBm or lower, so a gain of > 90dB
may be needed. What happens when the input signal is 0 dBm???
In gaining up the signal, we have to keep the noise and distortion
small relative to the signal power in order to meet the required
SNDR.
SDR Concept (minimize RF gain): ADC spec of > 100dB of
dynamic range (> 17bit) with a sampling frequency of 2fRF .
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Receiver Goal #2: Variable Amplification
LNA
VGA
ADC
Since the received power can vary greatly in dynamic range from
very weak levels (-110 dBm) to fairly strong signals (0 dBm), the
receiver should ideally have variable gain of ∼ 0-100 dB.
Without variable gain, the dynamic range of a receiver is limited
since the detector or ADC may have a limited range. For an ADC,
it’s roughly 6 dB/bit, and the power consumption grows
exponentially with the number of bits.
Variable gain at RF is difficult. But even if implemented, the
sensitivity would be limited by the presence of interfering signals.
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Receiver Goal #3: Freq. Translation
LNA
VGA
ADC
As the carrier frequency and the information signal are at very
disparate frequencies (say 1 GHz versus 1 MHz), we require
modulation and demodulation.
Also, we prefer to work at lower frequencies to save power.
We would like to frequency translate our signal to “baseband”
and perform filtering/gain rather than at RF. This means we
should “mix” the signal as soon as possible.
We shall see that mixers are prone to frequency translate
many different frequencies to the same “IF”, and so they are
relatively noisy and require image rejection (NF ∼ 10 dB). We
must precede the mixers with a low noise amplifier (LNA) to
overcome this noise.
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Receiver Goals #4: Filtering
LNA
VGA
ADC
Imagine trying to receive a signal at a power of -100 dBm in
the presence of an inband“jammer” or interference signal with
power -40 dBm.
We would like to set the gain at 100 dB, but this would
severly compress the receiver due to the jammer.
We must therefore apply a sharp filter to remove the jamming
signal before we apply all the gain.
If these jammers (blockers/interferers) are not attenuated,
they tend to reduce the gain of the signal (P−1dB ), increase
the noise figure of the receiver (through mixing noise in other
bands to the same IF, especially phase noise), and produce
intermodulation products that fall in band and reduce the
sensitivity of a receiver
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Filtering in Receivers
ADC
LNA
VGA
VCO
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Receiver Summary
+
-
LNA
ADC
I
ADC
Q
+
LO_I
+
-
LO_Q
+
High gain required to meet sensitivity (∼ 70dB)
DC offset cancellation is a must (due to mismatches)
Gain control to accommodate strong blockers
High linearity filtering stages to knock down blockers
Image rejection and harmonic rejection (cannot be filtered
easily)
Tunable baseband IF corner frequency (200 kHz 100 MHz)
for different standards
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Complex Baseband Representation
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Complex Baseband
Any passband waveform can be written in the following form:
sp (t) = a(t) cos [ωc t + θ(t)]
sp (t) = a(t) cos ωc t cos θ(t) − a(t) sin ωc t sin θ(t)
√
√
sp (t) = 2sc (t) cos ωc t − 2ss (t) sin ωc t
sc (t) , a(t) cos θ(t) = I (t)
ss (t) , a(t) sin θ(t) = Q(t)
q
a(t) = |s(t)| = sc2 (t) + ss2 (t)
θ(t) = tan−1
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ss (t)
sc (t)
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Complex Baseband (cont)
We define the complex baseband signal and show that all
operations at passband have a simple equivalent at complex
baseband:
s(t) = sc (t) + jss (t) = I (t) + jQ(t)
o
n√
2s(t)e jωc t
sp (t) = Re
||s||2 = ||sp ||2
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Orthogonality
An important relationship is the orthogonality between the
modulated I and Q signals: < xc , xs >=< Xc , Xs >= 0
This can be proved as follows (Parseval’s Relation):
Z ∞
< Xc , Xs >=
Xc (f )Xs∗ (f )df
xc (t) =
√
−∞
2sc (t) cos ωc t
xs (t) =
√
2ss (t) sin ωc t
1
xc (t) = √ (sc (t)e jωc t + sc (t)e −jωc t )
2
1
Xc (f ) = √ (Sc (f − fc ) + Sc (f + fc ))
2
1
Xs (f ) = √ (Ss (f − fc ) − Ss (f + fc ))
2j
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Orthogonality
< Xc , Xs >=
1
2j
Z
∞
−∞
((Sc (f − fc ) + Sc (f + fc )) × (Ss∗ (f − fc ) − Ss∗ (f + fc ))) df
In the above integral, if the carrier frequency is larger than the
signal bandwidth, then the frequency shifted signals do not
overlap
Sc (f − fc )Ss∗ (f + fc ) ≡ 0
< Xc , Xs >=
1
2j
Z
< Xc , Xs >=
∞
−∞
1
2j
Sc (f + fc )Ss∗ (f − fc ) ≡ 0
Sc (f − fc )Ss∗ (f − fc )df −
Z
∞
−∞
Sc (f )Ss∗ (f )df −
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Z
∞
−∞
Z
∞
−∞
Sc (f + fc )Ss∗ (f + fc )df
Sc (f )Ss∗ (f )df
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=0
Complex Baseband Spectrum
Due to this orthogonality, we can double the bandwidth of our
signal my modulating the I and Q independently. Also, we
have
< up , vp >=< uc , vc > + < us , vs >= Re [< u, v >]
Since the passband signal is real, it has a conjugate symmetric
spectrum about the origin. Let’s define the positive portion as
follows:
Sp+ (f ) = Sp (f )u(f )
Then the spectrum of the passband and baseband complex
signal are related by:
√
∗ (−f −f )
c
√
S(f ) = 2Sp+ (f + fc ) Sp (f ) = S(f −fc )+S
2
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Proof
Proof:
√
2s(t)e jωc t
√
V (f ) = 2S(f − fc )
v (t) =
v (t) + v (t)∗
2
∗
V (f ) + V (−f )
S(f − fc ) + S ∗ (−f − fc )
√
Sp (f ) =
=
2
2
Sp (t) = Re(v (t)) =
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The Image Problem
LNA
LO
IF
IF
IF
1
mr (t) (cos(2ωLO + ωIF )t + cos(ωIF )t)
2
1
mi (t) cos(ωLO −ωIF )t×cos(ωLO )t = mi (t) (cos(2ωLO − ωIF )t + cos(ωIF )t)
2
mr (t) cos(ωLO +ωIF )t×cos(ωLO )t =
After low-pass filtering the mixer output, the IF is given by
1
IFoutput = (mi (t) + mr (t)) cos(ωIF )t
2
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Image Problem (Freq Dom)
Positive Freq Exp Modulation
e+jωt
e+jωt
−LO
−IF
LO
IF
Negative Freq Exp Modulation
e−jωt
−LO
e−jωt
−IF
IF
LO
Cosine Modulation
−LO
−IF
IF
LO
Complex modulation shifts in only one direction . . . real
modulation shifts up and down
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Superheterodyne Architecture
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Superheterodyne
Off chip
Image
Filter
IF
Filter
LNA
Passive
BPF
The choice of the IF frequency dictated by:
If the IF is set too low, then we require a very high-Q image
reject filter, which introduces more loss and therefore higher
noise figure in the receiver (not to mention cost).
If the IF is set too high, then subsequent stages consume more
power (VGA and filters)
Typical IF frequency is 100-200 MHz.
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LO Planning in Superhet
Off chip
Image
Filter
IF
Filter
I
LNA
LO2
Passive
BPF
PLL2
LO1
Q
PLL1
Two separate VCOs and synthesizers are used. The IF LO is fixed,
while the RF LO is variable to down-convert the desired channel to
the passband of the IF filter (SAW). This typically results in a 3-4
chip solution with many off-chip components.
LO1 should never be made close to an integer multiple of LO2 for
any channel. The Nth harmonic of the the fixed LO2 could leak into
the RF mixer and cause unwanted mixing.
LO2
LO1
IF
n LO2
IF
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nLO2 leaks
into RF mixer
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The 1/2 IF Problem
LO
½ IF
IF
IF
Assume that there is a blocker half-way between the LO and the
desired channel. Due to second-order non-linearity in the RF
front-end:
2
1
1 + cos(2ωLO + ωIF )t
2
(t)
mblocker (t) cos(ωLO + ωIF )t = mblocker
2
2
If the LO has a second-order component, then this signal will fold
right on top of the desired signal at IF:
2
2
mblocker (t) cos(2ωLO + ωIF )t cos(2ωLO )t = mblocker
(t) cos(ωIF )t + · · ·
Note: Bandwidth expansion of blocker due to squaring operation.
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Half-IF Continued
DC
DC
½ IF
½ IF
IF
IF
If the IF stage has strong second-order non-linearity, then the
half-IF problem occurs through this mechanism:
2
1
2
2
2 mblocker (t) cos( ωIF )t = mblocker
(t) + mblocker
(t) cos(ωIF )t
2
This highlights the importance of frequency planning. One
should select the IF by making sure that there is no strong
half-IF blocker. If one exists, then the second-order
non-linearity must be carefully managed.
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Secondary Image Issue
DC IF2
IM2 LO2 IF1
LO1
IM2’
RF
When we down-convert twice using a mixer, we have to make
sure that the image in both the image bands is suppressed.
The secondary mixer has an image that may fall in a band
close to our desired signal, making the second image rejection
difficult.
IM2 = IF1 − 2IF2 = IF1 − 2(IF1 − LO2 ) = −IF1 + 2LO2
IM20 = IM2 + LO1 = LO1 + 2LO2 − IF1
= LO1 + 2LO2 − (RF − LO1 ) = 2(LO1 + LO2 ) − RF
Note LO1 + LO2 = RF − IF2
IM20 = 2RF − 2IF2 − RF = RF − 2IF2
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Dual-Conversion Single-Quad
Off chip
Image
Filter
IF
Filter
I
LNA
LO2
Passive
BPF
PLL2
LO1
Q
PLL1
Disadvantages:
Requires bulky off-chip SAW filters
As before, two synthesizers are required
Typically a three chip solution (RF, IF, and Synth)
Advantages:
Robust. The clear choice for extremely high sensitivity radios
High dynamic range SAW filter reduces/relaxes burden on
active circuits. This makes it much easier to design the active
circuitry.
By the same token, the power consumption is lower
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Image Reject Architectures
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Complex Mixer
+
+
y = x·z = (xr +jxi )·(zr +jzi )
–
+
+
+
yi = xi zr + xr zi
yr = xr zr − xi zi
y (t) = e jω0 t x(t)
Y (ω) = X (ω − ω0 )
A complex mixer is derived by simple substitution.
Note that a complex exponential only introduces a frequency
shift in one direction (no image rejection problems).
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Hilbert Architecture
A
LNA
+
90◦
IF
C
B
90◦
Image suppression by proper phase shifting.
RF = mr (t) cos(ωLO + ωIF )t + mi (t) cos(ωLO − ωIF )t
A = RF ×cos(ωLO t) =
1
1
mr (t) (cos(2ωLO + ωIF )t + cos(ωIF )t)+ mi (t) (cos(2ωLO − ωIF )t + cos(ωIF )t)
2
2
B = RF ×sin(ωLO t) =
1
1
mr (t) (sin(2ωLO + ωIF )t − sin(ωIF )t)+ mi (t) (sin(2ωLO − ωIF )t + sin(ωIF )t)
2
2
C =
1
1
mr (t) (− cos(2ωLO + ωIF )t + cos(ωIF )t) + mi (t) (− cos(2ωLO − ωIF )t − cos(ωIF )t)
2
2
IF
+
= A + C = mr (t) cos(ωIF t)
IF
−
= A − C = mi (t) cos(ωIF t)
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Sine/Cosine Together
Cosine Modulation
−LO
−IF
LO
IF
Sine Modulation
/j
/j
/j
/j
−LO
−IF
IF
LO
Delayed Sine Modulation
−LO
−IF
IF
LO
Since the sine treats positive/negative frequencies differently
(above/below LO), we can exploit this behavior
A 90◦ phase shift is needed to eliminate the image
90◦ phase shift equivalent to multiply by −jsign(f )
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Hilbert Implementation
Advantages:
Remove the external image-reject SAW filter
Better integration
Requires extremely good matching of components (paths
gain/phase). Without trimming/calibration, only ∼40dB
image rejection is possible. Many applications require 60dB or
more.
Power hungry (more mixers and higher cap loading).
A
Note: A real
implementation uses
45◦ /135◦ phase
shifters for better
matching/tracking.
45◦
D
45◦
+
LNA
135◦
C
B
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135◦
IF
Gain/Phase Imbalance
A = RF × (1 + α) cos(ωLO t +
B = RF × (1 − α) sin(ωLO t −
φ
2
φ
2
+ ωIF t − φ
) − sin(ωIF t −
2
− ωIF t − φ
) + sin(ωIF t −
2
)=
1 m (t)(1 − α) sin(2ω
LO t
2 r
1 m (t)(1 − α) sin(2ω
t
i
LO
2
1 m (t)(1 − α) − cos(2ω
LO t
2 r
1 m (t)(1 − α) − cos(2ω
LO t
2 i
IF = A + C =
+ ωIF t + φ
) + cos(ωIF t − φ
) +
2
2 − ωIF t + φ
) + cos(ωIF t + φ
)
2
2
1 m (t)(1 + α) cos(2ω
LO t
2 r
1 m (t)(1 + α) cos(2ω
LO t
2 i
C =
)=
mr (t)
φ
)
2 φ
)
2
+
+ ωIF t − φ
) + cos(ωIF t − φ
) +
2
2 − ωIF t − φ
) − cos(ωIF t − φ
)
2
2
(1 + α) cos(ωIF t −
φ
) + (1 − α) cos(ωIF t +
φ
+
2
2
2
φ
mi (t)
φ
(1 + α) cos(ωIF t + ) − (1 − α) cos(ωIF t − )
2
2
2
φ
φ
φ
φ
IF = mr (t) cos(ωIF t) cos( ) − α sin(ωIF t) sin( ) + mi (t) α cos(ωIF t) cos( ) − sin(ωIF t) sin( )
2
2
2
2
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)
Image-Reject Ratio
10
IR(dB) = 10 log
cos
φ
2
α cos
− α sin
φ
2
+ sin
φ
2
φ
2
!2
8
6
α2 + φ2
IR ≈=
4
4
2
0
0
1
2
3
4
Level of image rejection depends on amplitude/phase
mismatch
Typical op-chip values of 30-40 dB achieved (< 5◦ , < 0.6dB)
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RF/IF Phase Shift, Fixed LO
A
90◦
–
RF
LNA
A’
+
IF
90◦
RF’
B
RF = mr (t) cos(ωLO + ωIF )t + mi (t) cos(ωLO − ωIF )t
0
RF = −mr (t) sin(ωLO + ωIF )t − mi (t) sin(ωLO − ωIF )t
A = RF ×cos(ωLO t) =
0
B = RF ×cos(ωLO t) =
1
2 mr (t) (cos(2ωLO
0
ALPF
+ ωIF )t + cos(ωIF )t)+ 12 mi (t) (cos(2ωLO − ωIF )t + cos(ωIF )t)
= −mr (t) sin(ωIF t) − mi (t) sin(ωIF t)
1
2 mr (t) (− sin(2ωLO
IF
+
+ ωIF )t + sin(ωIF )t)+ 12 mi (t) (− sin(2ωLO − ωIF )t − sin(ωIF )t
0
= B − A = mr (t) sin(ωIF t)
This requires a 90◦ phase shift across the band. It’s much
easier to shift the phase of a single frequency (LO). Even
though the LO is variable, it’s a narrowband signal.
Polyphase filters can be used to do this, but a broadband
implementation requires many stages (high loss)
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Weaver Architecture
A
LNA
C
+
RF
90◦
LO1
90◦
B
+
LO2
–
IF
D
RF = mr (t) cos(ωLO1 + ωIF1 )t + mi (t) cos(ωLO1 − ωIF1 )t
IF1 = LO1 − RF
IF = LO2 − IF1 = LO2 − LO1 + RF = RF − (LO1 − LO2 )
Eliminates the need for a phase shift in the signal path. Easier to implement phase shift in
the LO path.
Can use a pair of quadrature VCOs. Requires 4X mixers!
Sensitive to second image.
mr
mi
ALPF = cos ωLO1 t × RF =
cos(ωIF1 )t +
cos(ωIF1 )t
2
2
BLPF = sin ωLO1 t × RF = −
CLPF = A × cos ωLO2 t =
mi
mr
sin(ωIF1 )t +
sin(ωIF1 )t
2
2
mr
mi
cos(ωIF )t +
cos(ωIF )t
4
4
DLPF = B × sin ωLO2 t = −
IF = C − D =
mr
mi
cos(ωIF )t +
cos(ωIF )t
4
4
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mr
cos ωIF t
2
Direct Conversion Architecture
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Direct Conversion (Zero-IF)
LO
DC
RF
LNA
ωRF = ωLO = ω0
DC
1
mr (t) cos(ωRF )t × cos(ωLO )t = mr (t) (1 + cos(2ω0 )t)
2
The most obvious choice of LO is the RF frequency, right?
IF = LO − RF = DC ?
Why not?
Even though the signal is its own image, if a complex
modulation is used, the complex envelope is asymmetric and
thus there is a “mangling” of the signal
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Direct Conversion (cont)
I
LNA
90◦
Q
Use orthogonal mixing to prevent signal folding and retain
both I and Q for complex demodulation (e.g. QPSK or QAM)
Since the image and the signal are the same, the image-reject
requirements are relaxed (it’s an SNR hit, so typically 20-25
dB is adequate)
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Problems with Zero-IF
LO = p(t) cos(ωLO t + φ(t))
2
2
LO×LO = p(t) +p(t) cos(2ωLO t+2φ(t))
I
DC
LNA
90◦
Dynamic DC
Offset
Q
Self-mixing of the LO signal is a big concern.
LO self-mixing degrades the SNR. The signal that reflects
from the antenna and is gained up appears at the input of the
mixer and mixes down to DC.
If the reflected signal varies in time, say due to a changing
VSWR on the antenna, then the DC offset is time-varying
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DC Offset
60 dB Gain
I
LNA
90◦
Q
1mV Offset
1V Offset
DC offsets that appear at the baseband experience a large
gain. This signal can easily saturate out the receive chain.
A large AC coupling capacitor or a programmable DC-offset
cancellation loop is required. The HPF corner should be low
(kHz), which requires a large capacitor.
Any transients require a large settling time as a result.
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Sensitivity to 2nd Order Disto
Assume two jammers have a frequency separation of ∆f :
s1 = m1 (t) cos(ω1 t)
s2 = m2 (t) cos(ω1 t + ∆ωt)
(s1 + s2 )2 = (m1 (t) cos ω1 t)2 + (m2 (t) cos ω2 t)2 +
2m1 (t)m2 (t) cos(ω1 t) cos(ω1 + ∆ω)t
LPF{(s1 + s2 )2 } = m1 (t)2 + m2 (t)2 + m1 (t)m2 (t) cos(∆ω)t
The two produce distortion at DC. The modulation of the
jammers gets doubled in bandwidth.
If the jammers are close together, then their inter-modulation
can also fall into the band of the receiver.
Even if it is out of band, it may be large enough to saturate
the receiver.
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Sensitivity to 1/f Noise
Since the IF is at DC, any low frequency noise, such as 1/f
noise, is particularly harmful.
CMOS has much higher 1/f noise, which requires careful
device sizing to ensure good operation.
Many cellular systems are narrowband and the entire
baseband may fall into the 1/f regime!
Example: GSM has a 200 kHz bandwidth. Suppose that the
flicker corner frequency is 100 kHz. The in-band noise
degradation is thus:
Noise
Density
2 =
vave
100 ni2
1/f Slope, 20dB/dec
1
200kHz
Z
100kHz
1kHz
a
df +
f
Z
200kHz
bdf
100kHz
a = 1kHz · 100 · vi2
b = vi2
ni2
200 kHz
100 kHz
10 kHz
1kHz
2 =
vave
1
100k
1
a ln
+ b(200k − 100k) =
(11.5a + 100kb) = 6.25vi2
200kHz
1k
200kHz
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Zero-IF Summary
DC offset and flicker noise a major concern. In addition to the
dominant mechanisms described earlier, other source of DC
offset include:
Strong blocker signal can leak into mixers and self-mix
Non 50% duty cycle of LO can cause output DC offset
Second order distortion a major concern and requires linear
LNA/mixer.
Despite these limitations, the zero-IF has many advantages,
such as elimination of the image rejection problem and the
elimination of all external SAW filters. This has made it a
very popular choice.
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Low-IF and Double Conversion Architectures
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LNA
90◦
Complex Image
Reject Filter
Low IF Architecture
I
Q
Instead of going to DC, go a low IF, low enough so that the
IF circuitry and filters can be implemented on-chip, yet high
enough to avoid problems around DC (flicker noise, offsets,
etc). Typical IF is twice the signal bandwidth.
The image is rejected through a complex filter. Polyphase
filters are popular choices.
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Sliding IF
I
LNA
cos(ω1 t)
ω2 =
÷
ω1
N
Q
cos(ω2 t)
sin(ω2 t)
Designing a receiver with two separate frequency synthesizers
(PLL’s) that are not integer related requires double the
hardware and may suffer from frequency pulling (interaction).
Derive second LO by dividing first:
1
ωRF = ω1 + ω2 = ω1 1 +
N
This architecture is the sliding IF receiver, so called because
the IF frequency is not fixed but moves.
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Sliding IF (cont)
For N = 2 (“free” quadrature), we have ωRF = ω1 23 , so that
the first IF is given by
1
2
ωIF = ωRF − ω1 = ωRF 1 −
= ωRF
3
3
As the input frequency moves, the first IF “slides” along.
Interestingly the image band is not as wide as the RF band.
Since the IF scales as we move the input, the image band
compresses:
2
1
ωIM,1 = ωRF ,1 −2(ωRF ,1 −ωLO,1 ) = ωRF ,1 1 − 2(1 − ) = ωRF ,1
3
3
1
ωIM,2 = ωRF ,2
3
1
BImage = ωIM,2 − ωIM,1 = BRF
3
Niknejad
Advanced IC’s for Comm
Double-Conversion Double-Quad
C
A
D
+
+
–
LNA
90
I
◦
+
+
90◦
E
B
Q
+
F
÷
The dual-conversion double-quad architecture has the
advantage of de-sensitizing the receiver gain and phase
imbalance of the I and Q paths.
Niknejad
Advanced IC’s for Comm
Analysis of Double/Double
Assuming ideal quadrature and no gain errors:
RF = mr (t) cos(ωLO1 + ωLO2 + ωIF )t + mi (t) cos(ωLO1 + ωLO2 − ωIF )t+
A = LPF{RF × cos(ωLO1 t)} =
B = LPF{RF × sin(ωLO1 t)} =
1
mr (t) cos(ωLO2 + ωIF )t+
mi (t) cos(ωLO2 − ωIF )t
−mr (t) sin(ωLO2 + ωIF )t
−mi (t) sin(ωLO2 − ωIF )t
2
1
2
1
C = LPF{A × cos(ωLO2 t)} =
1
D = LPF{A × sin(ωLO2 t)} =
mr (t) cos(ωIF )t+
mi (t) cos(ωIF )t
mr (t) sin(ωIF )t+
−mi (t) sin(ωIF )t
mr (t) sin(ωIF )t+
−mi (t) sin(ωIF )t
2
1
E = LPF{B × cos(ωLO2 t)} =
F = LPF{B × sin(ωLO2 t)} =
2
1
2
2
−mr (t) cos(ωIF )t+
−mi (t) cos(ωIF )t
I = C − F = (mr (t) + mi (t)) cos(ωIF )t
Q = D + E = (mr (t) − mi (t)) sin(ωIF )t
Niknejad
Advanced IC’s for Comm
Gain Error Analysis
RF = mr (t) cos(ωLO1 + ωLO2 + ωIF )t + mi (t) cos(ωLO1 + ωLO2 − ωIF )t+
∆a1
1
∆a1
mr (t) cos(ωLO2 + ωIF )t+
1+
cos(ωLO1 t)} =
1+
mi (t) cos(ωLO2 − ωIF )t
2
2
2
∆a1
1
∆a1
−mr (t) sin(ωLO2 + ωIF )t
B = LPF{RF × 1 −
sin(ωLO1 t)} =
1−
−mi (t) sin(ωLO2 − ωIF )t
2
2
2
∆a2
1
∆a1
∆a2
mr (t) cos(ωIF )t+
C = LPF{A × 1 +
cos(ωLO2 t)} =
1+
1+
mi (t) cos(ωIF )t
2
2
2
2
∆a2
1
∆a1
∆a2
mr (t) sin(ωIF )t+
D = LPF{A × 1 −
sin(ωLO2 t)} =
1+
1−
−mi (t) sin(ωIF )t
2
2
2
2
1
∆a1
∆a2
∆a2
mr (t) sin(ωIF )t+
cos(ωLO2 t)} =
1−
1+
E = LPF{B × 1 +
−mi (t) sin(ωIF )t
2
2
2
2
1
∆a1
∆a2
∆a2
−mr (t) cos(ωIF )t+
sin(ωLO2 t)} =
1−
1−
F = LPF{B × 1 −
−mi (t) cos(ωIF )t
2
2
2
2
A = LPF{RF ×
I = C − F = (1 + ∆a1 ∆a2 )(mr (t) + mi (t)) cos(ωIF )t
Q = D + E = (1 − ∆a1 ∆a2 )(mr (t) − mi (t)) sin(ωIF )t
The gain mismatch is reduced due to the product of two small
numbers (amplitude errors).
Niknejad
Advanced IC’s for Comm
Phase Error Analysis
RF = mr (t) cos(ωLO1 + ωLO2 + ωIF )t + mi (t) cos(ωLO1 + ωLO2 − ωIF )t+
1
mr (t) cos(ωLO2 + ωIF + φ1 )t+
A = LPF{RF × cos(ωLO1 t + φ1 )} =
mi (t) cos(ωLO2 − ωIF + φ1 )t
2
1
−mr (t) sin(ωLO2 + ωIF − φ1 )t
B = LPF{RF × sin(ωLO1 t − φ1 )} =
−mi (t) sin(ωLO2 − ωIF − +φ1 )t
2
1
mr (t) cos(ωIF + φ1 + φ2 )t+
C = LPF{A × cos(ωLO2 t + φ2 )} =
mi (t) cos(ωIF + φ1 + φ2 )t
2
1
mr (t) sin(ωIF − φ1 − φ2 )t+
D = LPF{A × sin(ωLO2 t − φ2 )} =
−mi (t) sin(ωIF − φ1 − φ2 )t
2
1
mr (t) sin(ωIF + φ1 + φ2 )t+
E = LPF{B × cos(ωLO2 t + φ2 )} =
−mi (t) sin(ωIF + φ1 + φ2 )t
2
1
−mr (t) cos(ωIF − φ1 − φ2 )t+
F = LPF{B × sin(ωLO2 t − φ2 )} =
−mi (t) cos(ωIF − φ1 − φ2 )t
2
I = C − F = (mr (t) + mi (t)) cos(φ1 + φ2 ) cos(ωIF t)
Q = D + E = (mr (t) − mi (t)) cos(φ1 + φ2 ) sin(ωIF )t
!
(φ1 + φ2 )2
cos(φ1 + φ2 ) ≈ 1 −
2
The phase error impacts the I/Q channels in the same way,
and as long as the phase errors are small, it has a minimal
impact on the gain of the I/Q channels.
Niknejad
Advanced IC’s for Comm
Double-Quad Low-IF
A
B
+
+
–
LNA
I
90◦
+
+
C
Q
+
90◦
D
Essentially a complex mixer topology. Mix RF I/Q with LO
I/Q to form baseband I/Q
Improved image rejection due to desensitization to quadrature
gain and phase error.
Niknejad
Advanced IC’s for Comm
References
1
Fundamental of Digital Communication, U. Madhow,
Cambridge 2008 O. Shanaa, EECS 290C Course Notes, 2005.
2
A.A. Abidi, “Radio frequency integrated circuits for portable
communications,” Proc. of CICC, pp. 151-158, May 1994.
3
A.A. Abidi, “Direct conversion radio transceivers for digital
communications,” Proc. of ISSCC, pp. 186-187, Feb. 1995.
4
J. Crols and M. Steyaert, “A single-chip 900MHz receiver
front-end with high performance low-IF topology,” IEEE
JSSC, vol. 30, no. 12, pp. 1483-1492, Dec. 1995.
5
B. Razavi, RF Microelectronics, Prentice Halls, 1998. J. Crols,
M. Steyaert, CMOS Wireless Transceiver Design, Kluwer
Academic Publishers, 1997.
6
Practical RF System Design, Willian F. Egdan, Wiley-IEEE
Press, 2003.
Niknejad
Advanced IC’s for Comm