A novel spectral subtraction technique for cognitive radios

Wendy Van Moer‡∗, Kurt Barbé‡ and Niclas Björsell∗

†

‡Vrije Universiteit Brussel, Dept. ELEC\M2ESA, Pleinlaan 2, B-1050 Brussels, Belgium ∗ Center for RF Measurement Technology, University of Gävle, 801 76 Gävle, Sweden

† Communications Systems Lab (CoS), Royal Institute of Technology (KTH), Stockholm, Sweden Email:

Abstract – In the past a lot of work has been done to

remove noise from speech. Most of the presented

techniques were derived from Boll’s spectral subtraction

technique. Roughly speaking the spectral subtraction

techniques consists of estimating the noise power during the periods when no speech is present and subtracting this

estimate of the noise power from the signal when speech is

present. This spectral subtraction technique could be a very

good in-band de-noising technique for communication

signals measured by cognitive radios. However, there is

one major drawback: one can never turn off the spectrum

so that no communication signals are present.

This paper presents an extended version of the spectral subtraction technique which does not require ‘speech free’

periods, but can determine the noise power from the empty

frequency bins in the spectrum. The presented method is

based on an autoregressive (AR) model, which is linear in

the parameters. Simulation results show that the presented technique is as performing as the original spectral

subtraction techniques without the need to turn off the

signals.

Keywords – cognitive radio, de-noising, spectral

subtraction, auto-regressive model

I. INTRODUCTION

Cognitive or ‘smart’ radios are able to decide for

themselves on which frequency to transmit by sensing the

spectrum for empty frequency bins [1]. The concept of

cognitive radio was first proposed in [2] and it has opened

up a new way of utilizing scarce wireless spectrum

resources. Using CR enables flexible access to the wireless

spectrum, which can improve the efficiency in spectrum

utilization significantly. For example, CR can be used in

‘empty’ frequencies in the TV broadcasting frequency

range, so called TV white space (TVWS), where one

method to find the white spaces is to use spectrum sensing [3]. The spectrum sensing device must then be able to

detect signal with a signal-to-noise ratio of down to -20 dB.

Reducing the noise is hence very important. Out-of-band

noise can easily be reduced by employing adaptive band

pass filters. In-band noise, however, can never be reduced

by classical filters and requires more involved techniques.

In the past, a lot of work has been done to remove noise

from speech signals. Most of the developed techniques

were based on the spectral subtraction technique presented

in [4]. The basic idea behind this method is to get an

estimate of the in-band noise power from measurements that were performed during speechless periods. This

estimate of the noise power will then be used to subtract it

from the speech signal. Over the years, other researchers

have extended this spectral subtraction technique in order

to get rid of the ‘musical noise’ [5] that is introduced by the

method, to be able to take into account varying signal-to-

noise ratios [6], colored noise [7],…

At first sight, it would be very beneficial to reuse the spectral subtraction method for cognitive radio

applications, where the detected signals can be very weak.

However, the spectral subtraction technique has one major

drawback: one needs to be able to turn off the signal for a

certain period of time. And this, however, is not possible in

the case of cognitive radio applications. One cannot turn off

the spectrum. This means that a different approach is

needed to find an estimate of the noise present on the

signal, or to separate the noise from the signal contribution.

In this paper we will present an extended spectral

subtraction technique. The proposed method is based on the idea of spectral subtraction, but it overcomes the need to

turn off the signal. Based on a recently developed spectrum

sensing technique [8], we will be able to split up the

measured spectrum in two groups: one containing ‘signal

bins’ and one containing ‘noise bins’. However, the signal

bins will be corrupted by colored noise, which we want to

subtract from the signal. Based on an autoregressive model,

we are able to get an estimate of the noise power from the

‘noise bins’. Or in other words, we will find an estimate of

the noise contribution on the signal lines from the noise

bins in the neighborhood of the signal lines.

Since we are not dealing in the application of cognitive

radio with white Gaussian noise, but with colored (filtered)

noise, one can argue that an autoregressive approach (AR)

will never be sufficient. An autoregressive moving average

(ARMA) approach is needed. However, by using Wold’s

method [11], which states that when dealing with stable

filters (all poles are located in the left half plane), one can

reduce the ARMA to an AR approach. The AR approach

has significant advantages: it is linear in the parameters,

does not require any iteration and a least squares solution is

the optimal one.

Section II describes the idea behind the spectral subtraction technique as well as the advantages and disadvantages.

Section III will propose a method to extend the spectral

subtraction technique for cognitive radios. In section IV

simulations will prove the capabilities of the developed

method.

II. SPECTRAL SUBTRACTION

The original idea behind spectral subtraction is quite

straight forward and can be found in [4]. This section will

give a short overview of the method.

A. Signal model and noise assumptions

Assume that a noise corrupted speech signal consists

of the summation of a noise-free speech signal and a

colored noise contribution :

(1)

Note that the noise is considered to be uncorrelated

with the signal .

is a noise process such that its power spectral density

is uniformly bounded for , with the

noise bandwidth. A stable filter exists where ∗ and denotes the underlying band limited

( ) white Gaussian noise.

Taking the Fourier transform of equation (1) at the angular

frequency results in:

(2)

where , and are the Fourier transform

coefficients of respectively , and .

B. Boll’s spectral subtraction

An estimate of the magnitude of the noise spectrum can be obtained by using its average value taken during speech free periods of the signal. The phase

estimation of the noise is set equal to the phase

of .

As a result, an estimate of the noise free signal is obtained as:

(3)

Or

With

and }.

This estimator results in a spectral error which generates some auditory effects. A lot of techniques have been

developed to attenuate these effects [5], [6], [7], but all

presented techniques assume that speech free periods are

available.

III. EXTENDING SPECTRAL SUBTRACTION

FOR COGNITIVE RADIOS

In order to use the spectral subtraction technique for cognitive radios, the assumption of having speech free

periods needs to be circumvented. Based on a recently

developed technique of spectrum sensing [8], [9], we will

present a new extended spectral subtraction method for

cognitive radios.

A. Spectrum sensing

A spectrum sensing technique is commonly used in

cognitive radios and tries to find white spaces in the

spectrum. Or in other words, it tries to find those bins in the spectrum where only noise is present. The spectrum

sensing technique used in this work is based on a

discriminant analysis algorithm. The main philosophy of

discriminant analysis is to partition the data in two groups:

one group of frequency lines that only contain noise and

one group of frequency lines that contain signal and noise.

Frequency lines that contain only noise are Rayleigh

distributed while the frequency lines containing signal and

noise are Rice distributed (since we are dealing with

amplitude data). The partitioning of the data into two

groups is done such that the groups are maximally

separated under the constraint that the variance within every group is as small as possible. Or in other words, the

technique tries to separate the Rice distributed spectral lines

from the Rayleigh distributed lines.

B. Autoregressive moving average model

Consider the following equation:

(4)

which represents the discrete Fourier coefficients of the

measured spectrum at frequencies

where

is the number of samples, the sample frequency and

denotes the number of the measured bin. represents

the noise free spectrum and is the spectrum of the

underlying band limited white Gaussian noise process

which follows a circular complex Gaussian distribution. As

can be seen from equation (4), we assume that the noise

free spectral components are disturbed by colored

noise

. Hence, we are working in an output

error framework [10].

Based on our spectrum sensing method, we were able to

split up the spectrum in two groups: one containing noise-

only lines and one containing signal lines.

For the spectral lines belonging to the noise group , equation (4) can be written as

(5)

since for . Hence, equation (5) shows

that the underlying time domain signal can be

represented by an autoregressive moving average (ARMA)

process [10]. Note however, that the white Gaussian noise

is unknown due to the fact that we are not measuring

transfer functions, but spectra. We need to find an estimate

of the noise filter

which turns the white noise

into colored noise, or in other words we are interested in

the power spectrum of the noise. If we know the power

spectrum of the noise, we can subtract it from the signal

lines that contain noise, as is requested by the spectral subtraction technique.

However, estimating the noise filter

requires an

estimate of which results in a moving average

model. This model is however, nonlinear in the parameters

and requires an iterative algorithm, which is unacceptable

for real-time implementation.

C. Autoregressive model

For stable filters, i.e. all poles in the left half plane, the

theorem of Wold [11] states that equation (5) can be

approximated by

(6)

This theorem is valid when the number of poles of

is

much higher than the sum of the number of poles and zeros

of

. As a result, the ARMA process of equation (5),

is now reduced to an autoregressive (AR) process, which is

linear in the parameters and does not require any iterative

algorithm. In other words, an AR approximation of the

colored noise is obtained.

D. Least squares estimator

In a next step, an estimation of

will be obtained from

the measured noise spectra with . The most optimal solution for a problem that is linear in the

parameters is obtained through a least squares solution.

By using a least squares estimator to estimate

from

the measured noise spectrum , the following cost

function is obtained:

K

(7)

since

(8)

Note that equation (8) is only valid for those frequency

lines belonging to . A bias will be introduced when

this is used for frequency lines not belonging to as can be seen from the following reasoning.

By taking the Wold’s theorem into account, the spectral

lines containing signal can be approximated by

(9)

With Multiplying left and right side of

equation (9) by results in,

Hence, a bias is introduced when inverse filtering the signal

lines. Instead, a compensation algorithm based on the noise

of neighboring lines will be used in the following

paragraph.

E. Compensation

Instead of using equation (7) to inverse filter the signal

lines, an interpolation technique will be used based on the estimation of the noise contribution on neighboring lines.

We will work under the assumption that between two

signal lines, at least one noise line is present.

Consider the signal line and its two direct

neighboring lines . An estimation

of the white noise on the signal line can be obtained through:

(10)

As a result, an estimation of the colored noise in the signal

line is obtained as:

(11)

Using equation (11) to subtract the noise from the signal

lines results in the following compensated spectrum

(12)

for . Note that

since

for white Gaussian noise.

Compensating the noise lines ( , results in

almost zero values ( .

IV. SIMULATION RESULTS

In order to prove the performance of the proposed extended

spectral subtraction technique, two test cases will be

simulated. In the first test, the colored noise on the

spectrum is simulated through an AR noise filter. In a

second test case, an ARMA noise filter will be used.

A. Case I: AR noise filter

For the first test, we start from an ideal noise free spectrum

that contains 39 spectral components (red dots in Figure 1).

Colored noise is added to this ideal spectrum by using an AR noise filter that contains two poles (blue line in Figure

1) and results in the noisy spectrum represented by the

green crosses in Figure 1. The signal-to-noise ratio of this

spectrum is -5 dB. Based on this noisy spectrum the noise

filter is estimated by a least squares estimator as in equation

(7) and is represented by the magenta curve in Figure 1.

The model of the noise filter consists of an AR model with

two poles. This noise filter is then used to compensate the

noisy spectrum as is done in equations (11) and (12).

From Figure 1, we can see from the compensated spectrum

at the non-excited lines (black crosses), that the noise level has dropped, compared to the non compensated spectrum

(green crosses). This is also reflected at the excited lines

(red starts) due to the fact that the compensated spectrum is

less scattered around the true spectrum (red dots). A gain in

signal-to-noise ratio after compensation of 8 dB is

obtained. This corresponds to the performance of the Boll

spectral subtraction technique [4].

B. Case II: ARMA noise filter

In the second test, we start from the same ideal noise free

spectrum as in case I (red dots in Figure 2). However, the

colored noise is now added to this ideal spectrum by using

an ARMA noise filter (blue line in Figure 2) that contains

four poles and 2 zeros. The resulting noisy spectrum is

represented by the green crosses in Figure 2. The signal-to-

noise ratio of this spectrum is -20 dB. Based on this noisy

spectrum, the noise filter is estimated by a least squares

estimator as in equation (7) and is represented by the magenta curve in Figure 2. The model of the noise filter

consists of an AR model with four poles. This noise filter is

then used to compensate the noisy spectrum as is done in

equations (11) and (12).

From Figure 2, we can see from the compensated spectrum

at the non-excited lines (black crosses), that the noise level

has dropped, compared to the non compensated spectrum

(green crosses). This is also reflected at the excited lines

(red starts) due to the fact that the compensated spectrum is

less scattered around the true spectrum (red dots). A gain in

signal-to-noise ratio after compensation of 8 dB is obtained, which corresponds to the performance of the Boll

spectral subtraction technique [4].

CONCLUSION

In this paper, we presented an extended spectral subtraction

technique based on an AR model which can be used for

cognitive radio. In contrast to earlier developed techniques,

the presented method does not require speech-free periods

and can be implemented real-time. The technique is based on an estimation of the noise power on the non-excited

frequency bins. The obtained gain in signal-to-noise ratio is

comparable to the performance of existing techniques.

ACKNOWLEDGEMENT

This research was funded by a postdoctoral fellowship of

the Research Foundation-Flanders (FWO).

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[3] FCC-10-174, FCC Second Memorandum and Order, September

23, 2010

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Figure 2: ARMA noise filter: ideal spectrum (red triangles), noise disturbed spectrum (green crosses), compensated spectrum

(black crosses for the noise lines, red stars for the signal lines),

real noise filter (blue line), estimated noise filter (magenta line).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-60

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Figure 1: AR noise filter: ideal spectrum (red triangles), noise disturbed spectrum (green crosses), compensated spectrum

(black crosses for the noise lines, red stars for the signal lines),

real noise filter (blue line), estimated noise filter (magenta line).

0 0.2 0.4 0.6 0.8 1-60

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