Investigation of an Optimal Utilization of Ultra-Wide Band Measurements for Position Purposes

Department for Information Technology

Vishnu Vardhan Siripi

Investigation of an Optimal Utilization of Ultra-Wide Band

Measurements for Position Purposes

Electrical Engineering Master Thesis

Date/Term: 2006-04-06 Supervisor: Dr. A. Jakobsson Examiner: Dr. A. Jakobsson

ABSTRACT

Ultra wideband (UWB) communication systems refers to systems whose bandwidth is many times greater than the “narrowband” systems (refers to a signal which occupies only small amount of space on the radio spectrum). UWB can be used for indoor, communications for high data rates, or very low data rates for substantial link distances because of the extremely large bandwidth, immune to multi-path fading, penetrations through concrete block or obstacles. UWB can also used for short distance ranging whose applications include asset location in a warehouse, position location for wireless sensor networks, and collision avoidance.

In order to verify analytical and simulation results with real-world measurements, the need for experimental UWB systems arises. The Institute of Communications Engineering [IANT] has developed a low-cost experimental UWB positioning system to test UWB based positioning concepts. The mobile devices use the avalanche effect of transistors for simple generation of bi-phase pulses and are TDMA multi-user capable.

The receiver is implemented in software and employs coherent cross-correlation with peak detection to localize the mobile unit via Time-Difference-Of-Arrival (TDOA) algorithms. Since the power of a proposed UWB system’s signal spread over a very wide bandwidth, the frequencies allocated to multiple existing narrowband systems may interfere with UWB spectrum. The goal of the filters discussed in this project is to cancel or suppress the interference while not distort the desired signal. To investigate the interference, we develop a algorithm to calculate the interference tones. In this thesis, we assume the interference to be narrowband interference (NBI) modeled as sinusoidal tones with unknown amplitude, frequency and phase. If we known the interference tones then it may be removed using a simple notched filter. Herein, we chose an adaptive filter so that it can adjust the interference tone automatically and cancel. In this thesis I tested adaptive filter technique to cancel interference cancellation (ie) LMS algorithm and Adaptive Noise Cancellation (ANC) technique. In this thesis performance of the both filters are compared.

Acknowledgement

Over the past years I’ve been inspired and encouraged by a lot of people around me.

Especially my supervisors, Dr. J. Schroeder and Dr A. Jakobsson – supervisors like you are an inspiration to any student. It has been a great pleasure working with you.

I wish to express my deepest gratitude to Dr. J. Schroeder for his unconditional support and guidance. My supervisor generosity and encouragement have not only enabled me to initiate and complete this project, but also enabled me to gain a clear understanding of the project.

I am grateful to the staff of the IANT for giving me a place to work and supporting me. I would like to thank Prof. Dr–Ing. K. Kyamakya and Dr. J. Schroeder for their considerable help on many areas during this project.

Many sincere thanks to Dr A. Jakobsson who helped me in completing my project report and also in thesis presentation.

Finally, I offer my heartfelt thanks to you all at Karlstad University for inviting me into a truly inspiring and relaxed environment. A special thanks to my greatest source of inspiration and encouragement — Mom and Dad. Thank you for everything!

Words alone cannot express my gratitude.

TABLE OF CONTENTS

CHAPTER 1

1.1 Introduction..………..7

1.2 Ultra-Short Pulse Generation Theory..………..8

1.3 Positioning………...11

1.3.1 Direct Calculation Method………13

1.3.2 Optimization Based Methods………15

1.4 Thesis Outline………..18

CHAPTER 2 2.1 Ultra-Wideband (UWB) Local Positioning System………19

2.2 Narrowband Interference Cancellation………20

2.3 Filter Design……….21

2.3.1 Adaptive Noise Canceling (ANC) Filter………...21

CHAPTER 3 3.1 Performance……….35

3.1.1 Laboratory recorded data………..35

3.2 Comparing with other filter methods.……….……….42

CHAPTER 4 4.1 Comparing the Performance of Different Filters……….40

4.2 Comparing the mean, standard deviation error.………...……....44

CHAPTER 5 Conclusions………48

REFERENCE……….49

APPENDIX………51

LIST OF FIGURES

Figure 1.1 FCC Definition of UWB..………..7

Figure 1.2 FCC General UWB Emission Limits..………...8

Figure 1.3 Gaussian Pulse...……….9

Figure 1.4 Gaussian Pulse First Derivative..………9

Figure 1.5 Gaussian Pulse Second Derivative…...………..9

Figure 1.6 Gaussian Modulated RF Pulse ……….10

Figure 1.7 BPAM pulse shapes for ‘1’ and ‘0’ bits………...11

Figure 1.8 PPM pulse shapes for ‘1’ and ‘0’ bits………..11

Figure 1.9 Examples of the pulse waveform used for PSM modulation………...11

Figure 1.10 OOK pulse shapes for ‘1’ and ‘0’ bits..………..11

Figure 1.11 General Position System……….12

Figure 2.1 Block Diagram of Adaptive Noise Cancellation Filter (ANC)………21

Figure 2.2 (a) Block diagram of the new model of the adaptive system (b) Transfer-function….………...23

Figure 2.3 Block Diagram Representation of ANC….………...23

Figure 2.4 Plots of N β function for various N….………...26

Figure 3.1 Coherent Cross-Correlation Receiver………...32

Figure 3.2 Power spectrum of the interference tones…………..………..33

Figure 3.3 (a) Position estimation using ANC Filter series # 1 (b) Position estimation error using ANC Filter.………..…….35

Figure 3.4 (a) Position estimation using ANC Filter series # 2 (b) Position estimation error using ANC Filter.…………..……….36

Figure 3.5 (a) Position estimation using band pass filter with pass bands up to1.1 GHz. (b) Position estimation error using band pass filter with pass bands Up to1.1 GHz………...37

Figure 3.6 (a) Position estimation using LMS Filter (b) Position estimation error using LMS Filter………39

Figure 4.1 (a) Mean of each measurement of ANC Filter (b) Standard Deviation of each measurement of ANC Filter………...40

Figure 4.2 (a) Mean of each measurement of band pass filter

(b) Standard Deviation of each measurement of band pass Filter……... ……41 Figure 4.3 (a) Mean of each measurement of LMS Filter1

(b) Standard Deviation of each measurement of LMS Filter………42 Figure 4.4 (a) Mean Error of each measurement of ANC Filter

(b) Standard Deviation Error of each measurement of ANC Filter…...43 Figure 4.5 (a) Mean Error of each measurement of band pass Filter

(b) Standard Deviation Error of each measurement of band pass Filter …...44 Figure 4.6 (a) Mean Error of each measurement of LMS Filter

(b) Standard Deviation Error of each measurement of LMS Filter………...45 Table 1………...46 Table 2………...47

CHAPTER 1

1.1 Introduction

The Federal Communications Commission (FCC) approved a First Report and Order allowing the production and operation of unlicensed ultra-wideband (UWB) devices in February 2002 [1]. The report specified application areas and provided corresponding operating frequency ranges and power limitations. The application areas included vehicular radar systems, communications and measurement systems, imaging systems and position applications. Of all the systems mentioned, a position application is the focus of this thesis.

What is UWB?

The accepted definition of Ultra Wide-Band (UWB) devices proposed by FCC in February 2002 [1], is a signal with fractional bandwidth greater than 0.20 or which occupies more than 500 MHz of the spectrum, i.e.,

Fractional Bandwidth = ²

(

_f^f_H^H₊⁻ _f^f_L^L

)

(1.1) where

• fH upper frequencies of the bandwidth

• f^L lower frequencies of the bandwidth

Figure 1.1 below provides an illustration comparing the fractional bandwidth of a narrowband signal and a UWB signal, with f denoting the signal’s center frequency. _c

Figure 1.1 FCC Definition of UWB [1]

Figure 1.2 demonstrates that a UWB signal’s bandwidth can cover a large range of frequencies. It is therefore important that UWB devices use a low transmit power spectral density in order to not interfere with existing narrowband communications systems. For this reason the FCC has provided a preliminary “conservative” spectral mask for all UWB systems.

Figure 1.2 FCC General UWB Emission Limits [1]

1.2 Ultra-Short Pulse Generation Theory

UWB signals may be generated by a great variety of methods. The most popular pulse shape for UWB communication system is the Gaussian pulse or a derivative of the Gaussian pulse due to mathematical convenience and ease of generation. The Gaussian pulse isdefined as

2) 2 2 ( ) (

2 2

) 1

( ^{μ σ}

πσ

= e ^t−

t

p

− ∞ < t < ∞

(1.2)

where σ is the standard deviation of the Gaussian pulse in seconds, and μ is the location in time for the midpoint of the Gaussian pulse in seconds, and the pulse width t is _p related to the standard deviation as t_p =2πσ . It is also of interest to consider the derivatives of the Gaussian pulse since most wideband antennas may differentiate the generated pulse (assumed to be Gaussian) with respect to time. Antennas are an extremely important component of a UWB system and a great deal of consideration

should be given to this area when designing a UWB system. The first derivative and second derivative of a Gaussian pulse are given by

)2 6 (

' 32

)

( k te ^kt

t

p ⎟⎟ ⁻

⎠

⎞

⎜⎜

⎝

=⎛

π ^(1.3)

)2 2 (

41

'' 32 2 (1 2( ) ) ) 9

( k kt e ^kt

t

p _⎟^⎟ − ⁻

⎠

⎞

⎜⎜

⎝

=⎛

π ^(1.4)

where K is a constant that determines the pulse width and we assumed μ = 0.

Figures 1.3 - 1.5 shows as example a Gaussian as well as the first and second derivative of the pulse. The time axis is arbitrary and depends on the values assumed above for k and σ.

Figure 1.3 Gaussian Pulse [2] Figure 1.4 Gaussian Pulse First Derivative [2]

Figure 1.5 Gaussian Pulse Second Derivative [2]

The FCC has issued UWB emission limits in the form of a spectral mask for indoor and outdoor systems as shown in figure 1.2 [1]. In the band from 3.1 GHz to 10.6 GHz, UWB can use the FCC Part 15 rules with a peak value of −41 dBm/MHz. The first derivative of the Gaussian pulse does not meet the FCC requirement no matter what value of the pulse width is used. Therefore, a new pulse shape i.e., sinusoid using a second derivative Gaussian pulse is used which satisfies the FCC emission requirements. So we modulate a sinusoid using a second derivative Gaussian pulse, shifting the pulse into the proper frequency range, which is defined as

) 2 cos(

1 ) 1

( ^{( )2}

2 41

2 2

8 f t

t

p ^kt _c

k f

e e

k

c

π _π π

− +

⎟⎠

⎜ ⎞

⎝

=⎛ (1.5)

where

f c

is the center frequency of the RF pulse. Figure 1.6 gives an example of the resulting RF pulse.

Figure 1.6 Gaussian Modulated RF Pulse [2]

UWB systems are based on impulse radio concepts because impulse radio resolves multipath as well as low implementation complexity [3]. Although numerous modulation techniques are used with impulse-radio based UWB concepts, four common schemes are used (1) Pulse Position Modulation (PPM), (2) On-Off Keying (OOK) Modulation, (3) Pulse Amplitude Modulation (PAM) and (4) Pulse Shape Modulation (PSM). In the case of PPM, time instant 0 represents a bit ‘0’ and a bit ‘1’ is represented by shifting the pulse relative to time by the amount ofδ . In case of PAM, "1" positive pulse is transmitted, whereas a "0" negative pulse is transmitted. In case of PSM, different orthogonal waveforms are used to represent bit ‘0’ and ‘1’. In case of OOK, where a pulse is transmitted to represent a binary “1”, while no pulse is transmitted for a binary

hopping (TH-PPM) techniques due to their simplicity and flexibility. The modulation techniques are shown in the Figures 1.7-1.10.

UWB system with this technique and operating at very low RF power levels have demonstrated good performance for short- and long-range data links, positioning measurements accurate to within a few centimetres, and high-performance through-wall motion sensing radars.

Figure 1.7 BPAM modulation [4] Figure 1.8 PPM modulations [4]

Figure 1.9 PSM modulation [4] Figure 1.10 OOK modulation [4]

1.3 Positioning:

If the time of arrival of a pulse is known with little uncertainty, then it is possible to estimate the distance traveled by the pulse from the source. It is possible to use triangulation techniques to estimate the position of the source by combining the distance estimates at multiple receivers. For UWB systems with potential bandwidth of 7.5 GHz, the maximum time resolution of a pulse is of the order of 133 ps, so when a pulse arrives, it is possible to know within 133 ps. For modest bandwidth of 500 MHz, the corresponding time resolution is 2 ns, which corresponds to approximately 60 cm. With any UWB signal, it is possible to achieve sub-meter accuracy in positioning.

There are many position estimation techniques such as Angle-of-Arrival(AOA), time measurements (Time-of-Arrival (TOA), Time-of-Flight, Time difference of Arrival (TDOA)). Figure 1.11 shows a general positioning system configuration with base stations or sensors and the device to be located. The most common approach to estimate the position is directly solving a set of simultaneous equations based on the TDOA measurements. With three sensors for 2-D (two dimensional) and four sensors for 3-D (three dimensional) positioning exact solutions can be obtained. For an over determined system (with redundant sensors), Taylor series expansion may be used to produce a linearized, least-square solution iteratively to the position estimate. To estimate position we choose direct calculation method and the non-linear optimization algorithm.

Figure 1.11 General Position System [4]

1.3.1 Direct Calculation Method:

Methods for calculating the TDOA and mobile position have been reviewed previously [5]. Some methods calculate the two dimensional position and others the three dimensional position depending on the degree of simplicity desired. Here, we examine a set of equations needed to locate the three dimensional position of a mobile. Here, for easy calculations we considered 4 sensors in calculating the TDOA measurements. The essence of the TDOA technique is the equation for the distance between sensor i and the device is [4]

(x−x_i)² +(y−y_i)² +(z−z_i)² =c(t_i−t₀) i=1,2,3,4 (1.6)

where (x, y, z) and (

x

_i,

y

_i,

z

_i) are the coordinates of the device and the i^th sensors, c is the speed of light,

t

_i is the signal TOA at sensor i and t₀ is the unknown transmit time at the device. Here, for simplicity we ignore the difference between the true and the measured TOAs. We need to solve for the three unknowns x, y and z to estimate the position of the device. Squaring the equation (1.6) on both sides we get

0 2 2

2 2

2 ( ) ( ) ( )

)

(x−x_i + y− y_i + z−z_i =c t_i−t i=1,2,3,4 (1.7) Subtracting (1.7) for i=1 from (1.7) for i=2,3,4 produces

(

^{x x y y z z}

)

t t t c

t c

ct _{i i i i}

i i 1 1 1 1

1 1

0 2 2 2

) (

2 ) 1 2 (

1 − − −

+ −

−

=

β

i=2,3,4 (1.8)

where

) (

₁² ₁² ₁²

2 2 2 1

1 1

1 1

1 1

z y x z y x

z z z

y y y

x x x

i i i i

i i

i i

i i

+ +

− + +

=

−

=

−

=

−

=

β

Defining the TDOA between sensors i and j as j

i

ij

t t

t = −

Δ

Eliminating

t

0 and solving the equations we get [4]

B Az

x = +

(1.9)

D Cz

y = +

(1.10)

G I G

H G

z = − H ±

_⎟⎟

−

⎠

⎜⎜ ⎞

⎝

⎛ ²

2

2

^(1.11)

where

1 2 2 1

1 2 2 1

b a b a

c b c A b

−

= −

1 2 2 1

2 1 1 2

b a b a

g b g B b

−

= −

1 2 2 1

2 1 1 2

b a b a

c a c C a

−

= −

1 2 2 1

1 2 2 1

b a b a

g a g D a

−

= −

2 1

2

2 + − +

=

A C E

G

) )

( ) (

(

2

A B x

1

C D y

1

z

1

EF

H

= − + − − −

2 2 1 2 1 2

1) ( )

(B x D y z F

I = − + − + −

) 1 (

21 21

21 12

z C y A t x

E c + +

= Δ (2( ) )

2 1

2 ^{21 21 21}

12

12 + −

β

+ Δ

= Δ x B y D

t c t

F c

21 13 31

12

1 t x t x

a =Δ −Δ a₂ =Δt₁₂x₄₁−Δt₁₄x₂₁

21 13 31

12

1 t y t y

b =Δ −Δ b₂ =Δt₁₂y₄₁−Δt₁₄y₂₁

21 13 31

12

1 t z t z

c =Δ −Δ c₂ =Δt₁₂z₄₁−Δt₁₄z₂₁

) 2 (

1

21 13 31

12 32

13 12 2

1

c t t t t β t β

g = Δ Δ Δ + Δ − Δ

) 2 (

1

21 14 41

12 42

14 12 2

2

c t t t t β t β

g = Δ Δ Δ + Δ − Δ

The two estimated z [equation 1.11] values are then substituted in equations (1.9, 1.10) to produce the coordinates x and y. However, there is only one desirable solution. We can

dismiss one of the solution if it beyond the monitored area. If both solutions are reasonable and they are very close, we may choose the average as position estimate.

Otherwise, an ambiguity occurs. If both solutions are beyond the monitored area, we may add a fifth sensor. Then we have five different combinations, producing five different results.

1.3.2 Optimization Based Methods:

Objective Function:

The objective function is defined as sum of the squared range errors of all sensors to obtain accurate position estimate, defines as [4]

∑

=

= ^N

i fi x y z t t

z y x F

1

2( , , , ) 2

) 1 , , ,

( _{0 0} (1.12)

where

• N number of active sensors/base stations

• (x,y,z) unknown position coordinates

• t ₀ unknown transmit time

• f_i (x,y,z,t₀)= (x−x_i)² +(y−y_i)² +(z−z_i)² −c(t_i −t₀)

where

t

_iis the estimated TOA at the ith sensor and c is the speed of light. The optimization purpose is to minimize this objective function to produce the optimal position estimate. For notational simplicity, we define

p=(

x , y , z , t

₀)^T

f(p)=(f₁(p),f₂(p),KK,f_N(p))^T Then, the objective function becomes

f(p) ²

2 ) 1 p ( =

f (1.13)

Expanding the objective function in the Taylor series at the current point p and taking _k first three terms, we have [4]

F p_k s_k F p_k g^T_ks_k s_k^TG(p_k)s_k 2

) 1 ( )

( + ≈ + + (1.14)

where s is the directional vector (or increment vector) to be determined, _k G(p_k) is the Hessian of the objective function and g_k is a vector of the first partial derivatives of the objective function at p is [4] _k

gk =∇f(x,y,z,t₀) p=pk

T

k k

k

k t p p

p f z p p f y p p f x p f

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ∂ =

= ∂

∂

= ∂

∂

= ∂

∂

= ∂

0

, ,

,

Minimizing the equation (1.14) yields

G(p_k)s_k =−g_k (1.15)

The minimization in which s is defined by equation (1.15) is Newton’s method [6]. For _k minimizing function s the recurrence relation is [4] _k

p_k+1 = p_k − (G(p_k))⁻¹J_s(p_k ) (1.16)

where J is the Jacobian matrix of _s s i.e., _k

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ∂ =

= ∂

∂

= ∂

∂

= ∂

∂

= ∂

k k

k

k y p p z p p t p p

p x p

J

s s s s

s 0

, ,

,

Using Newton’s method we have J_s =2*J_k f(p_k) and

( ) 2* *( ) 2* ( ) ² ( )

1

p f p f J

J p

G ^N _i

i i

k T k

k = +

∑

∇

= (1.17)

where ∇ denotes the Laplacian. ²

The calculation of the second order information in the Hessian is avoided, a simplified expression can be approached from equation (1.17) i.e., [4]

J^T_k J_ks_k =−J^T_k f(p_k) (1.18)

where J is the Jacobian matrix of f(P) at _k p defined as _k

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ∂ =

= ∂

∂

= ∂

∂

= ∂

∂

= ∂

k k

k

k k t p p

p f z p p f y p p f x p J f

0

, ,

,

This method is called the Gauss-Newton method. When J is full rank, we have the _k linear least-squares solution

^s_k ⁼⁻

(

^JT_k ^J_k

)

^−1J_k^{T f(p}_k^{) (1.19)}

When the second-order information in the Hessian is necessary, then we may use Levenberg-Marquardt method [7, 8] i.e.

(J_k^TJ_k +λI)s_k =−J^T_k f(p_k) (1.20)

where λ is a non-negative scalar that controls both the magnitude and direction of s . _k One way to find a position estimate from TDOA measurements is the direct solution of a set of pseudo-range equations, discussed in section 1.3.1 and in [9, 10]. Here, we used Bancroft’s algorithm [10] for solving the hyperbolic localization problem. Since the direct solution to hyperbolic positioning has poor performance when the standard deviation of the range errors increases, algorithms from nonlinear optimization theory are more appropriate for position estimation as discussed in section 1.3.2 and in [11, 12]. In the case of TDOA the objective function can be defined as the nonlinear least-square range difference errors of all N base stations [4]

∑ ∑

⁻

= =+

= ¹

1 1

2( ) )

( ^{N N}

i j ij

i

x f x

F (1.21)

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − − −

− Δ

= ^{∧ ∧}

2 2

)

( _{ij i j}

ij x c t x x x x

f (1.22)

where c is speed of light,

T

z y x

X ⎥⎦⎤

⎢⎣⎡

= ^{∧ ∧ ∧}

∧ , , is the position estimate vector X =

[

x,y,z

]

^Tis the real position vector, ^Xk =

[

^x_k,^y_k,^z_k

]

^T,^k∈ⁱ, ^j is the position vector of the base

station k, and Δ the time difference estimate between base station i and j. To find a t_ij position estimate, the objective function is minimized using a steepest descent quasi- Newton algorithm, which has been found to perform best in our scenario. The initial position has been chosen as the geometrical center of the room.

1.4 Thesis Outline

The thesis focuses on position applications for UWB systems. Chapter 2 discusses narrowband interference and filter technique to minimize narrowband interference.

Chapter 3 provides the comparison of the performance of different filters. Chapter 4 provides the comparison of the statistical properties of different filters. Chapter 5 provides conclusions and the direction of future work.

CHAPTER 2

Ultra-Wideband (UWB) for Local Positioning System

2.1 BACKGROUND/MOTIVATION

Services like 911 will permit a mobile telephone user to be located within 50 m when a call is placed to the emergency number. In commercial applications, more precise indoor geolocation technology will have the ability to:

• Track the elderly or children who are away from supervision.

• Locate portable equipment in hospitals.

• Provide information to track prison inmates.

• Provide navigation for the blind and other handicapped people.

• Provide navigation for police, fire fighters, and soldiers to safely complete rescue operations inside buildings.

• High accuracy, i.e., better than 1m and locations should be in 3 dimensions

• Good radio penetration through structures

• Tolerance of high levels of reflection

The Global Positioning System (GPS), a space-based radio navigation system already in use for outdoor services [13], does not work well in indoor environments, because the GPS signal is not strong enough to penetrate through most materials and when an object blocks the GPS satellite, the signal is corrupted. Attenuation and multipath reflections of the line-of-sight (LOS) signal (or direct path) by the walls, floors, and ceiling of a building are the main factors preventing typical GPS receivers from functioning indoors.

In dangerous or hostile situations, it is important to know each person’s position and physiological status in a building at all times. For example, in a fire or hostage situation, it is vital to determine not only a person’s location in the building but also the physical condition of the person. The indoor environment is usually cluttered with furniture’s, stock (in shops and warehouses), vehicles, and of course people. This clutter obstructs direct signals, and it reflects signal via indirect paths resulting in a longer path and add delay in receiving the signal. Based on this idea, a new technology called “ultra wide band (UWB)”, provides a possible solution to the problems encountered in indoor environments. UWB signals have been shown to be immune to multipath fading [14].

In order to verify analytical and simulation results with real-world measurements, the need for experimental UWB systems arises. The Institute of Communications Engineering [IANT] at the University of Hannover has developed a low-cost experimental UWB positioning system. We can find the detailed description of architecture and implementation of a low-cost experimental UWB local positioning system [15].

2.2 Narrowband Interference Cancellation:

There have been several research efforts investigating the impact of a UWB signal on existing systems. In [16], test results examining the effects of UWB signals on many different legacy military systems. The power of a proposed UWB system’s signal spread over a very wide bandwidth. We know a signal with a moderate bandwidth is spread over several gigahertz, so moderate power corresponds to a lower power spectral density. This appears as noise in a receiver. So the UWB transmissions may interfere with some devices. One of the biggest concerns is the impact of UWB on GPS since air traffic relies heavily on this technology.

The frequencies allocated to multiple existing narrowband systems may interfere with the UWB spectrum which hence will suffer. In a good system, this interference should be mitigated. This chapter provides some background information on narrowband interference cancellation. The majority of the techniques used for suppressing narrowband interference involve filtering. The goal of any of these filters is to cancel or suppress the interference while not distorting the desired signal. To investigate the interference, we assumed the interference as narrowband interference (NBI) according to [17] modeled as a sum of sinusoidal tones with unknown amplitude, frequency and phase defined as

∑ +

= = m

n n nt n

t

I( ) 1

α

cos(

ω φ

) ^(2.1)

where

m is the Number of interference tones,

α

_n,

ω

_n,

φ

_n are the amplitude, frequency, and Phase of the n^th sinusoid, respectively.

Our main task is to suppress the interference without loosing the information. If we know the sinusoidal interference, a fixed notch is implemented to eliminate the interference. If the interfering sinusoid drifts slowly in frequency then the fixed notch cannot work. In such situations, we use adaptive filtering. Two examples of applicable adaptive filters are

the LMS algorithm or RLS algorithm and other solution is adaptive noise canceling (ANC).

2.3 Filter Design:

2.3.1 Adaptive Noise Canceling (ANC) Filter:

We use a fixed notch filter tuned to the frequency of the interference to suppress a sinusoidal interference corrupting an information-bearing signal. If the notch is very narrow and the interfering sinusoid drifts slowly in frequency then the fixed notch cannot work. Clearly, this, calls for an adaptive solution. One such solution is provided by the use of adaptive noise canceling. Figure 2.1 shows the block diagram of a dual-input adaptive noise canceling. The primary input consists of information-bearing signal and sinusoidal interference. The reference input is sinusoidal interference. For the adaptive filter, we may use a transversal filter whose tap weights are adapted by means of the LMS algorithm. The filter uses the reference input to provide (at its output) an estimate of the sinusoidal interference signal contained in the primary input. Thus, by subtracting the adaptive filter output from the primary input, the sinusoidal interference is minimized. In particular, an adaptive noise canceller using LMS algorithm has two important characteristics

• The canceller behaves as an adaptive notch filter.

• The notch in the frequency response of the canceller can be made very sharp at precisely the frequency of the sinusoidal interference by choosing a small enough value for the step-size parameterμ.

Figure 2.1 Block Diagram of Adaptive Noise Cancellation System [18]

Adaptive Filter

∑ Primary Input

rk

Reference Input Ik

yk

+ -

Error Signal

ek

System Output

Here, the primary input consist of [19]

)

cos( ₀

0

ω θ

α

+

+

=s k

r_{k k r} (2.2)

where

sk is information-bearing signal,

α

0 is the amplitude of the sinusoidal interference,

ω

_r is the normalized angular frequency, and

θ

₀ is the phase. Similarly, the reference input consist of

I_k =

α

cos(

ω

_rk +

θ

) (2.3)

where

α

is the amplitude and

θ

is the phase of the sinusoidal interference which are allowed to be different from those in the primary input. We describe the tap-weight update by means of the equations [19]

∑⁻

= −

=

¹

0 M

i k i

T k

k

w I

y

^(2.4)

e

_k

= r

_k

− y

_k ^(2.5)

i k k k

k

w e I

w

₊₁

= + μ

₋ i=^0,1,...,M −¹ (2.6)

where M is the length of the transversal filter and μis the step-size. We restructure the block diagram of adaptive noise canceller with different set of inputs and outputs as shown in Figure 2.2. With this new representation, the sinusoidal input

I

k^{, the Nweights}

of the filter and the weight-updated equation of the LMS algorithm are all lumped together into a single (open-loop) system.

Figure 2.2 (a) Block diagram of the new model of the adaptive system (b) Transfer-function [18]

Figure 2.3 Block Diagram Representation of ANC System [18]

α + z⁻¹

∑

∑ 1

) 1

( = −

z z U

1 ,k+

Wi W _ik y ^ik

I ik

ek

-

y₁k

y Nk

yk

ek

rk

G(z )

+

Figure 2.3 is a block diagram representation of Figure 2.2b. Our task to find G(z). To do so, consider the

i

^th element of an reference input,

I

_ik

,

with arbitrary phase angle ,θ_i as shown in [18]

)

cos( _{r i}

ik kT

I =

α ω

+

θ

⎥

⎦

⎢ ⎤

⎣

⎡ +

=

α

e^jωrkTe^jθi e^{− jωrkT}e^{− jθi}

2 ^(2.7)

As seen in Figure 2.3, the input

I

_kis multiplied by the estimation error

e

_k implying that the z transform can be written as [18]

{ }

⎭⎬

⎫

⎩⎨ + ⎧

⎭⎬

⎫

⎩⎨

= e^{j i}z⎧e e^{j rkT} e^{− j i}z e e^{− j rkT} Ii

e

z _{k k}

α

^θ _k ^ω

α

^θ _k ^ω

2

2

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

=

α

e^jθiE⎛ze^{− jωrT} e^{− jθi}E ze^jωrT

2 ^(2.8)

Now taking the z transform to the updated tap-weight equation defined as [18]

zW_i(z)=W_i(z)+

μ

z(e_kI_ik) (2.9)

Substituting z{e_kI_ik} in the equation (2.9) and solving w_i(z) we get

⎥⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

= U z e^{j i}E⎛ze^{− j rT} e^{− j i}E ze^{j rT} z

W_i

μα

_{( )} ^{θ ω θ ω}

) 2

( (2.10)

where

1 ) 1

( = −

z z

U .

Next taking the z transform to the filter output

Y

kdefined in equation (2.4), [18]

Y_i(z)= z{w_ikI_ik}

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

=

α

W_i⎛ze^{− jωrT} e^jθi W_i ze^jωrT e^{− jθi} 2

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

=

μα

U₍ze^−jωrT₎ e^jθie^jθiE⎛ze^{− jωrT}e^{− jωrT} 4

2

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝

+

μα

U ze^−jωrT e^{− jθi}e^jθiE⎛ze^jωrTe^{− jωrT} )

4 (

2

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝

+

μα

U ze^jωrT e^{− jθi}e^jθiE⎛ze^jωrTe^{− jωrT} )

4 (

2

⎥

⎦

⎢ ⎤

⎣

+ ⎡

⎟

⎠

⎜ ⎞

⎝

⎛

^j _r^{T j} _r^T

j i j i rT

j e e E ze e

ze

U ^{ω θ θ ω ω}

μα

_{( )}

4

2

(2.11)

⎥⎦

⎢ ⎤

⎣ + ⎡

⎥⎦⎤

⎢⎣⎡ +

=

⎟

⎠

⎜ ⎞

⎝

⎛

− j rT j rT j rT − j i j rT

ze E e

ze U ze

U ze

U z E z

Y_i

μα

^{ω ω}

μα

^{ω 2 θ 2 ω}

) 4 (

) (

) (

) 4 (

)

( ^{2 2}

⎥

⎦

⎢ ⎤

⎣

+ ⎡

⎟

⎠

⎜ ⎞

⎝

⎛

⁻

−j rT e j iE ze j rT

ze

U ^{ω θ ω}

μα

_{( )} ^{2 2}

4

2

(2.12)

∑

=

= ^N

i Yi z z

Y

1

) ( )

(

⎥⎦⎤

⎢⎣⎡ +

= ( ) ( ⁻ ) ( )

4

2 j rT j rT

ze U ze

U z

N

μα

E ^{ω ω}

(2.13a)

⎥ ∑

⎦

⎢ ⎤

⎣ + ⎡

=

⎟

−

⎠

⎜ ⎞

⎝

⎛

^N

i

j i rT

T j

j r E ze e

ze U

1

2 2 2

) 4 (

ω θ

μα

ω

(2.13b)

⎥ ∑

⎦

⎢ ⎤

⎣ + ⎡

=

⎟ ⎠

⎜ ⎞

⎝

⎛

⁻

− ^N

i j i rT

T j

j r E ze e

ze U

1

2 2 2

) 4 (

ω θ

μα

ω

The terms (2.13a) represents the time-invariant (TI) component, which is independent of the phaseθ_i, where as the terms in (2.13b) represents time-varying components which depend on the phaseθ_i. Our goal is to have only G(z), so we have to remove the term

(2.13b). Since θ_i is the arbitrary phase shift of the ith element so θ_i =θ −ω_rT

[ ]

i−1 as shown in [18], i.e.,

) ,

)) (

1 ( 2

( 2

1

N rT e

e ^{j i j rT N}

N

i

ω

ω

β

θ

θ ± − −

± =

∑

=

(2.14)

where the function β is defined as

) sin(

) ) sin(

,

( T

T N N

T

r

r

ω

r

ω ω

β

=

This allows (2.13) to be approximated as [18]

( ) ( ) ( ) 4

) 4 (

2

2

μα

μα

^{ω ω} ₊

⎥⎦⎤

⎢⎣⎡ +

= N E z U ze^{− j rT} U ze^{j rT} z

Y

β

⁽

ω

_r^T,N)

[ ]

^TV (2.15)

The

[ ]

^TV in the equation (2.15) is eliminated by selecting the f_rT(ω_r =2πf_r)values between 0 and 0.5. One way to get f_rT between 0 and 0.5 is to decrease sampling rate 1/T or increase number of weights N in the adaptive filter. We can see in the figure 2.4, when

N

β is plotted as a function of f_rT(ω_r =2πf_r), for several values of N, ≈0 N

β as long as f_rT is not close to 0 or 0.5.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-1 0 1

ß/N

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-1 0 1

ß/N

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-0.5 0 0.5

fT

ß/N

N = 10 WEIGHTS

N = 20 WEIGHTS

N = 32 WEIGHTS

Figure 2.4 Plots of

N

β function for various N

If we assume that ≈0 N

β , then Y(z) can be approximated with only the time-invariant

component, i.e., [18]

⎥⎦⎤

⎢⎣⎡ +

= ( ) ( ⁻ ) ( )

) 4

( N ² E z U ze ^{j rT} U ze^{j rT} z

Y

μα

^{ω ω}

(2.16) The open-loop transfer function relating y(n) to e(n) is

⎥⎦⎤

⎢⎣⎡ +

=

= ( ⁻ ) ( )

4 )

( ) ) (

( N ² U ze ^{j rT} U ze^{j rT}

z E

z z Y

G

μα

^{ω ω}

where

1 ) 1

( = −

z z

U

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− +

−

= −

1 1 1

1 4

2

rT j rT

j ze

ze N

ω ω

μα

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+ +

−

−

= +

−

−

1 ) 2 (

2 ) (

4

2

rT j rT

j

rT j rT

j

e e

z z

e e

z N

ω ω

ω

μα

ω

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+

−

= −

1 ) cos(

2 2

1 ) cos(

2

2

rT z

z

rT N z

ω μα ω

(2.17)

The task is to find, closed-loop feedback system whose transfer function H(z) from d to _k

e

k is given by [18]

[ ]

1 ) cos(

2 2

1 ) 2 cos(

2

1

1 )

( 1

1 )

( ) ) (

(

+

−

− +

+ =

=

=

rT z

z

rT N z

z G z

D z z E

H

ω μα ω

⇒

⎟⎟

⎠

⎞

⎜⎜

⎝ +⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

− ⎛

=

−

−

+

−

1 4 ) 4 cos(

1

1 ) cos(

2

2 2 2

2

2 )

(

ω μα μα

ω T N N z

T z

z

r r

z z

H (2.18)

Equation (2.18) represents second-order digital notch filter with a notch at normalized angular frequency rω . The 3-dB bandwidth of the adaptive noise canceller is determined by considering the two half power points on the unit circle that are 2 times as far from the poles as they are from the zeros. Using this geometric, we find the 3-dB bandwidth is [19]

s T rad

BW N

2 μα 2

≈

(2.19)

Therefore, the smaller the

μ

, the smaller the bandwidth

BW

is, and the sharper the notch is.

Suppose the reference input is a sum of M sinusoids, i.e.,

∑

=

+

=

^M

m m m

k

kT

I

1

)

cos( ω θ

α

(2.20)

Then,

i

^th element of a input

i

-vector,

I

_ik can be written as [18]

∑

=

+

=

^M

m m im

ik

kT

I

1

) cos( ω θ

α

where

^θ

_im

^{= θ}

_m

^{− ω}

_m

^T [ ] ^{i − 1}

^(2.21)

From (2.8)-(2.15), [18]

∑

=

⎥⎦ ⎤

⎢⎣ ⎡ − +

=

^M

m

m m m

T ze j T U

ze j U z

N E z Y

1

2

( ) ( )

) 4 (

)

( μ α ω ω

∑

[ ]

≠

=

∑ = +

^M

−

m n m

M

n

n m n

m

T N TV

1

1

) 2 ,

4 (

ω β ω

α μα

[ ]

∑

=

∑ = +

^M

+

m M

n

n m n

m

T N TV

1

1

) 2 ,

4 (

ω β ω

α

μα

(2.22)

By decreasing T or increasing N, the number of weights in the adaptive filter would make 0

/ )]

2 ,

[( + ≈

N N

n T

m

ω

β ω

and the difference frequencies , )]/ 0

[( 2− ≈

N N nT

m ω

β ω .