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