Derivative-Based
Quadrature Identification of Channel Delays
Jinming Ge
Vaisala Inc
Louisville, CO 80027, USA
Abstract – Real-time detection of phase or time delay
between two ADC sample channels, especially when
fractional-delay filter is involved, often uses quadrature-
filter, which may not be cost-effective since two filter
channels have to be constructed to deal with each live
ADC sample stream. This paper presents a phase delay
quadrature detection method based on derivatives of the
sample stream. The challenge is to deal with the inherent
detection error of the naïve derivative when high
normalized frequency has to be used in RF/IF
applications. The cause of the error is analyzed, and a
proprietary algorithm is developed to cancel the error at
the critical quadrature crossing boundary, namely 0, ±90
and ±180 degree.
Keywords: phase delay, quadrature, derivative, fractional-
delay filter, real-time, cost-effective.
1 Introduction
The phase of a complex waveform described as
)1(A QjI +=∠φ
can often be obtained by
)2()/(atan IQ=φ
If the waveform is passing through a digital signal processing
(DSP) device: ADC sampled, filtered, two filters (I and Q)
have to be constructed, which may not be cost-effective in
some applications. Fig.1 illustrated a real application, in which
a waveform [1] is received from a radar front-end processing
unit, with wide dynamic range. Since the ADC doesn’t have
enough dynamic range to match the signal’s range, the signal
is “split” into two ADC channels, with one channel deals with
attenuated signal, so the overall system will not saturate when
input signal reaches its highest level. During the pre-
processing, the original signal maybe also has been phase
transformed (separated) in these two channels, besides the
intended gain separation. Before merging into an output signal
that ideally has the same characteristics of the original signal,
both the intended gain separation and unintended phase
separation must be corrected. The phase correction is done
through a reconfigurable fractional-delay finite impulse
response filter (RFDFIR) [2][3], together with a phase
sensitive detection (PSD) module. To construct two separate I
and Q filters for both the high and low gain channels will
significantly increase the implementation cost: the FPGA area
budget within a radar video processing (RVP) [4] device.
Fig 1. An example application where phase detection
with complex filters (I, Q) is not cost-effective.
A novel real-time detection of phase delay over ±180
degree range without using IQ filter is presented in the
following. Section 2 describes the challenge of using naïve
derivative method, the inherent quardarature detection error
when very high normalized frequency has to be used in radar
IF domain. A proprietary algorithm is used to counter the
naïve quadrature detection error by realizing that it is the
quadrature identification itself instead of the absolute error
affects the accuracy of overall phase detection. Section 3
presents a real-life application result and further discussions
are in Section 4.
2 Derivative-Based Detection
2.1 Fundamentals of Naïve Derivatives
To cover the full range, ±180 degree angle phase
(delay) detection, the quadrature information of the angle
can be derived from the cosine alone. When the angle
detected from asin term (which covers 0~±90 degree) is
known, the actual phase can be deduced as:
)3()asin180(:)asin180(?)0(sin:asin?)0(cos −−−>>=φ
Fig. 2 shows the identification of quadrature. Note that
only the sign of the cosine term, not necessary its accurate
value is needed to identify the quadrature correctly, as long
as the quadrature crossing critical points, namely the ±90
boundaries can be identified uniquely.
The cos term can be derived by using derivatives of
the incoming waveform stream, especially in baseband
sampling, where the normalized frequency is low, or
equivalently, the ADC sampling frequency is far higher
than signal frequency – as a result, many samples per cycle
can be sampled and used to calculate the derivative; or the
sampling period, T, is relatively very short, as defined
mathematically:
)4(,0,)sin()sin(
)cos( Tasttt
tttt ∆→∆
∆
−∆+=
But in RF/IF signal processing, quite often bandpass
sampling, where low sampling frequency is used. Even
processing at aliasing frequency, the normalized frequency
is still very high. Fig. 3 shows an example, where a
60MHz IF signal is sampled at 72MHz. Only 6 samples
per cycle can be obtained even at the relatively lower
aliasing frequency (12MHz). Therefore the assumption in
equation (4) is not valid and considerable error will be
resulted for the derivative, as shown in Fig. 4: the phase
error between the ideal derivative (when T is tiny, shown
in cyan) and the actual one (when T is corresponding to 60
degree, shown in red) is corresponding to about half of the
sampling period.
0 60 120 180 240 300 360-100
-50
0
50
100
aliasing wave phase (degree) -tick as sample clock
wave
ma
gnitude
Bandpass Sampling of 60 MHz IF with 72 MHz ADC
Fig. 3. Relative high normalized frequency often used in
bandpass RF/IF sampling, with only few samples per
cycle to be used for derivatives: 60MHz IF (in green),
12MHz (in blue) with dash-line represents sampled wave
while solid line for the analog wave, sample frequency as
72MHZ (every 60 degrees of the aliasing wave)
Fig 2. Quadrature Identification with cosine
2.2 Improved Derivative and Phase Detection
The derivative error is a function of sampling
frequency, as shown in Fig. 4, or more precisely, of
normalized frequency. It is also related to the relative
phase delay itself when used for phase/time delay
detection; in this case, both the sin and cos terms can be
deduced using cross correlation of the two channel waves,
as shown in Fig.1. The derivative error around the critical
boundary-crossing points (i.e. ±90 degree) can be reduced
by using a proprietary algorithm, as shown in Fig. 5.
As indicated in section 2.1, only the sign of cosine
term is used to identify whether the phase is to the left or
right of the qudrature plane (Fig. 2), not the absolute phase
(acos) value, so the results shown in Fig.5 is not surprising.
Although the acos value around the phase 0 and ±180
degree is far off from the actual (about 30 degree error),
but the quadrture (left/right) can still be correctly identified
based on the sign of cosine. For example, around phase
angle 0, the acos produces value as about 30 degree instead
of 0, but the sign of cos(0) and cos(30) are the same, i.e.
positive (+); around the phase ±180 degree, the acos
produces value as around 150 degree instead of 180, but
both have the same sign in terms of cos so they will not
affect the quadrature identification either. Around the
critical ±90 degree, where the sign of cos term is abruptly
switching, the derivative method produces smooth angle
transition, error nearly as zero, as clearly shown in Fig.5.
3 An Example of Derivative-based
Phase Delay Detection
As shown in Fig.1, waveforms from two channels can
have intended gain separation and unintended phase/time
delay separation [1]. The waveforms are shown in Fig.6.
Both the gain and phase delay can be detected and then
adjusted before merging into a wider dynamic range
0 60 120 180 240 300 360-100
-50
0
50
100
tick as sample clock
wave
ma
gni
tude
Phase Error: Derivative of ADC Samples
Fig 4. The challenge for phase detection from ADC wave
when bandpass sampling with high normalized
frequency. Ideal/ADC wave of the alias 12MHz (in blue
solid/dash); ideal derivative (cos) of 12MHz wave (in
cyan); actual derivative (in red dash) and fitting (in red
dot)
-180 -90 0 90 180-45
0
45
actual phase delay (degree)
phase
de
tecti
on e
rror
(deg
ree)
Phase Delay Detection Error of Two ADC Waves
Fig 5. The phase detection error: cos for quadrature
identification (in blue) and the overall phase detection
error (in cyan). Note that although the acos (in blue) can
have as high as 30 degree absolute error at non-critical 0
and 180 degree, it can still identify the quadrature
correctly since only the sign of cos is used.
0 0.5 1 1.5 2
x 104
-100
-80
-60
-40
-20
0
20
40
60
80
100
sample tick
bandlimited radar IF wave before ADCtime series samples
Fig 6. Synthetic radar waveforms of high and low gain
channel of Fig.1 with both gain and phase separations.
waveform.
When derivative-based phase detection is used, the
detected phase are used to generate a new set of FIR
coefficients for both high and low gain channel to make
the filtered output phase aligned before merging – a matter
of simple switch between these two channels to use only
unsaturated output from corresponding channel. One
criterion to judge the accuracy of both phase detection and
correction is the phase noise of the merged waveform –
ideally perfectly aligned. Fig. 7 shows the merged
waveform, with a general noise power (high gain channel
relative to low gain channel) of -60dB.
4 Further Discussions
The accuracy of phase delay detection is
fundamentally based on cross-correlation of two channels,
where the number of total correlated samples used will play
an important role, depending on the channel noise. This is
more important for the derivative-based cosine term
detection than the sine term itself, as seen from equation
(3) at the critical boundary crossing angle ±90 degree.
Besides using more correlated samples, in a closed-
loop system, more iteration may be used to remedy
inadequate accuracy of cosine term detection around the
critical point to control the system in a stable state.
5 References
[1] J. Ge and A. Siggia, Weather Radar Virtual Signal
Generator as Test Bench for Algorithm Development,
The 16th Symposium of Meteorological Observations
and Instrumentation, 92nd AMS Annual Meeting,
New Orleans, USA, 22-26 January 2012
[2] J. Ge, Model and Algorithm for Fractional Delay HPF,
The 2011 International Conference on Scientific
Computing, Las Vegas, USA, 18-21 July 2011
[3] T. Laakso, V. Valimaki, M. Karjalainen and U.
Laine, Splitting the Unit Delay, IEEE Signal
Processing Magzine, Jan., 1996.
[4] RVP900 User’s Manual, Vaisala, Feb., 2010
0 0.5 1 1.5 2
x 104
-1.5
-1
-0.5
0
0.5
1
1.5x 10
4
sample tick
merged wavetime series samples
Fig 7. The merged waveform using derivative-based
phase detection