FAST FREQUENCY SYNTHESIZER

CONCEPT BASED ON DIGITAL TUNING

AND I/Q SIGNAL PROCESSING

C. Caballero Gaudes 1, M. Valkama 1, M. Renfors 1, and J. Ajanki 2

1
Tampere University of Technology
P.O. Box 553, FIN-33101 Tampere, Finland
{caballer,valkama,mr}@cs.tut.fi
2
Satel Oy, P.O. Box 142, FIN-24101 Salo, Finland
jaakko.ajanki@satel.fi

DSP-2002, July 1-3, Santorini, Greece

 INTRODUCTION
– The design of frequency synthesizers is challenging for wireless applications:
→ spectral purity, high frequency range, fast tuning, power consumption, etc.
– In this paper, the idea of combining digital and analog synthesis
techniques for achieving these goals is discussed and analyzed
→ the proposed architecture uses I/Q modulation to translate a digitally
synthesized tuneable low frequency tone to the final frequency range
– Practical problems: unavoidable mismatches between the amplitudes and
phases of the I and Q branches result in imperfect sideband rejection
degrading the spectral purity of the synthesized signal.
– A compensation structure based on digital pre-distortion of the low frequency
tone is presented to enhance the signal quality
→ practical algorithms for updating the compensator parameters are proposed
based on minimizing the envelope variation of the synthesizer output signal
→ simulation results are also presented to illustrate the efficiency of the
proposed synthesizer concept

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 SYNTHESIZER ARCHITECTURE
Fixed
LO
Pre-Compensation ILO QLO
INCO D/A x1(t) xI (t)
a11 LPF

a21 + Output
NCO / DDS x(t)
a12 –

a22
QNCO D/A x2(t) xQ (t)
LPF

{a11 , a12 , a21 , a22}

Digital A(t) Envelope

A/D
Processing Detector

Figure 1: Synthesizer architecture where digital pre-compensation is used to compensate for the
non-idealities of the analog part.

Fast Frequency Synthesizer Concept based on Digital Tuning and I/Q Signal Processing 3 / 15
– Signal Analysis:
→ NCO signals: INCO(t) = cos(ωNCOt) and QNCO(t)= sin(ωNCOt)
→ I/Q mixer: ILO(t)=cos(ωLOt) and QLO(t)= gsin(ωLOt+ φ) where g and φ denote
the gain and phase imbalances of the mixing stage
→ furthermore, let θ1 and θ2 denote the relative phase shifts due to the D/A
converters and branch filters.
– Now, the synthesizer output signal x(t) can be easily shown to be of the form
x(t)= Re[xLP(t)exp(jωLOt)] where the lowpass equivalent xLP(t) is given by
xLP (t ) = αe jω NCOt + βe − jω NCOt (1)

– Thus, the synthesizer output consists of two spectral components:

→ the desired tone at ωLO + ωNCO and
→ the image tone at ωLO – ωNCO
whose relative strengts |α |2 and |β |2 depend on g, φ, and θ = θ2 – θ1, as well as
on the compensation parameters aij.

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– With practical analog electronics, the image tone attenuation defined here as
L = | α |2 | β |2 (2)

is only 20-40 dB if no compensation is used (i.e., a11 =a22 = 1 and a12 = a21 = 0)
→ in wireless systems, these levels of synthesizer spurious tones can result in
severe interchannel interference (ICI)
→ in general, image tone attenuations in the order of 50-80 dB are needed in
wireless applications
– The image tone at the synthesizer output can, however, be canceled with
proper pre-compensation parameters aij.
– In fact, setting β = 0 and α = exp(jθ1) will force the output to be a pure
sinusoidal (with relative phase θ1).
– These optimum parameters aij can be easily shown to be of the form
sin(θ ) cos(θ )
a11 = 1, a12 = tan(φ ), a 21 =− , a 22 = . (3)
g cos(φ ) g cos(φ )

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– If the only motivation is really just to cancel the image tone (i.e., to set β = 0), a
more simple solution is also available.
– Using the notation ψ = φ – θ, the solution can be formulated as
1
a11 = 1, a12 = tan(ψ ), a 21 = 0, a 22 = (4)
g cos(ψ )

for which also α ≈ exp(jθ1) with practical imbalance values.

– Thus, only two actual compensation parameters a12 and a22 need to be
implemented.
– Notice that only the phase error difference ψ = φ – θ and the gain mismatch g
are actually contributing to the image tone, and therefore, on the ideal
compensation parameters.

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 PRACTICAL UPDATE ALGORITHMS
– In practice, the imbalances g, φ, and θ needed to calculate the compensators of
(3) or (4) are unknown and need to be measured or estimated somehow.
– To emphasize implementation simplicity, the approach taken here is to carry
out the estimation using only the envelope

A(t ) = | xLP (t ) | = | αe jω NCOt + βe − jω NCOt |= | α |2 + | β |2 +2 Re[αβ *e j 2ω NCOt ] (5)

of the synthesizer output.

– As is obvious, the envelope is flat only if the image tone is completely
attenuated (β = 0).
– Thus, instead of estimating g and ψ, a more simple approach is to directly
adapt a12 and a22 to minimize the envelope variation.
– One possibility is to consider the envelope peak-to-peak (PP) value
d A = max{A(t )} − min{ A(t )}. (6)

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– Given that |α|> |β| which is always the case with practical imbalance values, the
result of (6) can also be expressed as

d A = 2 | β |= 1 + g 2 a22
2 2
+ a12 − 2 ga22 (cos(ψ ) + a12 sin(ψ )) . (7)

– Though not strictly parabolic, the dA -surface having a unique minimum lends
itself well to iterative minimization.
– The true gradient of dA (derivative with respect to a12 and a22) depends,
however, on g and ψ which are unknown.
– A practical approach is then to adapt only one parameter (either a12 or a22)
at a time.
– The direction (sign) of the needed one dimensional gradient at each iteration
can be determined based on observing the behaviour of dA between two
previous adaptations of the corresponding parameter.
– Assuming that a12 and a22 are updated at odd and even adaptation instants,
respectively, this leads to the following update rule

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 d A ( n)
aˆ12 (n) = aˆ12 (n − 2) + K 12 (n)λ1
λ 2 + d A ( n) (8a)
aˆ 22 (n) = aˆ 22 (n − 1)

for n odd with K12 ∈{+1,–1} and

aˆ12 (n) = aˆ12 (n − 1)
d A ( n) (8b)
aˆ 22 (n) = aˆ 22 (n − 2) + K 22 (n)λ1
λ 2 + d A ( n)

for n even with K22 ∈{+1,–1}.

– In the above adaptation scheme, K12(n)= K12(n–2) if dA(n–1)≤dA(n–2) and
K12(n)= –K12(n–2) if dA(n–1)> dA(n–2) with similar reasoning for K22.
– Step-size parameters λ1 >0 and λ2 ≥ 0 are used here to control the adaptation
speed and steady-state accuracy. Notice that the magnitude of the update term
is always between zero and λ1, depending on the value of dA.

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 EXAMPLE SIMULATIONS
– To demonstrate the proposed synthesizer concept with digital pre-
compensation, some computer simulations are carried out.
– In the simulations, imbalance values of g =1.05, φ = 6°, and θ = 1° (ψ = φ – θ = 5°)
are used, corresponding to an image tone attenuation of approximately 26 dB.
– Pre-compensation parameters are initialized as a12 =0 and a22 =1 and are then
iteratively updated one-by-one to minimize the peak envelope variation using
(8a) and (8b).
– In general, updating is carried out once per envelope cycle and step-size
values of λ1 =0.01 and λ2 =0.02 are used.
– With these example values, the dA -surface is illustrated in Figure 2 as a
function of a12 and a22.
– Also shown in the a12 ,a22 -plane is the behaviour of the pre-distortion
parameters during one simulation realization.

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 0.2

0.15
A
d

0.1

0.05

0
1.1
1 0.2
0.15
a 0.1
22 0.9 0.05
0 a12
0.8 −0.05
Figure 2: Peak envelope variation dA as a function of the compensation parameters a12 and a22
for g = 1.05 and ψ = 5°. Also shown in the a12 ,a22 -plane is one realization of the compensation
parameters during the adaptation.
Fast Frequency Synthesizer Concept based on Digital Tuning and I/Q Signal Processing 11 / 15
– The corresponding output envelope is depicted in Figure 3 verifying successful
synthesizer operation.
– With these imbalance and step-size values, the steady-state operation is
reached in 50 iterations or so and the steady-state image attenuation is in the
order of 100-150 dB.

Output Envelope
1.1
Amplitude

1.05

0.95
0 20 40 60 80 100
Time in Envelope Cycles

Figure 3: Envelope of the synthesizer output signal during the adaptation (g = 1.05 and ψ = 5°).

Fast Frequency Synthesizer Concept based on Digital Tuning and I/Q Signal Processing 12 / 15
– To illustrate the ability to perform the compensation also in time-varying
environments, an abrupt change in the imbalances is tested.
– The gain mismatch coefficient g is changed from 1.05 to 0.98 and the I/Q mixer
phase imbalance φ from 6° to 4°.
– The resulting output envelope is presented in Figure 4 evidencing fast and
accurate synthesizer operation also in case of time-varying imbalances.

Output Envelope
1.1

1.05
Amplitude

0.95

0.9
0 20 40 60 80 100
Time in Envelope Cycles
Figure 4: Envelope of the synthesizer output signal during the adaptation when the initial
imbalances g = 1.05 and ψ = 5° are changed to g = 0.98 and ψ = 3°, respectively.
Fast Frequency Synthesizer Concept based on Digital Tuning and I/Q Signal Processing 13 / 15
 SOME PRACTICAL MATTERS
– The output envelope is periodic with fundamental frequency 2ωNCO
→ if dA is determined and compensation parameters updated once per
envelope cycle (or once per several cycles), the frequency range of the
digital synthesis part should be selected in such a manner that the envelope
variation rate does not limit the whole synthesizer's settling time
→ on the other hand, higher frequencies in the digital part always imply higher
sampling rates and higher power consumption
→ proper compromise between these two issues is needed
– Data and coefficient wordlengths:
→ according to preliminary results, image tone attenuations in the order of 80
dB are achievable with 16 bits, more detailed analysis still needed
→ one important issue in this context is also the available accuracy of the
envelope variation measurements
– Local oscillator leakage: creates an additional spectral component at ωLO, can
be compensated by adding proper constants to the low frequency (NCO)
signals.
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 CONCLUSIONS
– In this contribution, frequency synthesizer design for wireless applications was
considered.
– To achieve fast switching capabilities with high operating frequencies and
reasonable power consumption, the approach was to use I/Q modulation of
digitally tunable low frequency tone.
– The practical imbalance problem with analog I/Q signal processing was then
considered and analyzed.
– Based on this analysis, a digital pre-compensation structure was presented
together with some simple yet efficient approaches to determine the
compensator coefficients.
– The proper operation of the whole synthesizer was demonstrated using
computer simulations both in time-invariant and time-varying situations.
– Future work should be directed to more detailed evaluation of the finite
wordlength effects (hardware prototyping, etc.).

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