SIGNAL COMPRESSION USING THE DISCRETE LINEAR CHIRP TRANSFORM (DLCT)

Osama A. Alkishriwo and Luis F. Chaparro

University of PittsburghDepartment of Electrical and Computer Engineering

1140 Benedum HallPittsburgh, PA, USA, 15261

ABSTRACTSignal compression aims to decrease transmission rate (in-crease storage capacity) by reducing the amount of data nec-essary to be transmitted. The discrete linear chirp transform(DLCT) is a joint frequency instantaneous-frequency trans-form that decomposes the signal in terms of linear chirps. TheDLCT can be used to transform signals that are not sparsein either time or frequency, such as linear chirps, into sparsesignals. In this paper, we propose a new algorithm for sig-nal compression based on the direct and the dual DLCT, de-pending on the sparsity of the signal in either time or in fre-quency. Furthermore, we develop a data structure for the ex-tracted coefficients of compressed signals. In the data struc-ture, the extracted parameters are arranged in certain waythat are predetermined for the compress and decompress pro-cesses. The ability of the proposed method in signal com-pression are demonstrated using test as well as actual signals.The results are compared with those obtained with compres-sive sensing (CS) method.

Index Terms— discrete linear chirp transform, signalcompression, compressive sensing, sparsity, duality

1. INTRODUCTION

The growth of communication systems and information tech-nology, and their ability to serve voice, image, and video,requires more data to be transmitted or stored. Signal com-pression transforms a signal into an efficient compact form,for transmission or storage, that can be decompressed backto produce a close approximation of the original signal. Thegoal of signal compression is to minimize data rate to con-serve bandwidth, while keeping the quality and intelligibilityof the original signal. Unfortunately, the compression ratiois inversely proportional to the quality of the signal. Hence,there is always a tradeoff between compression ratio and qual-ity [1, 2].

Compressive sensing (CS) [3] aims to take advantage ofthe signal’s sparser representation dictated by the uncertaintyprinciple. For instance, in [4] the signal to be compressedis represent in the sparser domain, using the discrete cosine

transform (DCT). Taking random measurements from thenew sparse signal so that the length of the measurement issmaller than the length of the original signal, the originalsignal can be reconstructed from the measurements using`1-optimization. Although CS provides very good results forsignals that are sparse in either time or frequency, it does notfor signals that are not significantly sparse in either time orfrequency domains such as the case of chirp signals [5],[6].Time-frequency analysis is needed to obtain an intermedi-ate domain where the signal is sparser than in time or infrequency. The Fractional Fourier Transform [8, 9] or thepolynomial time-frequency transforms [10] can be used, herewe propose a joint frequency instantaneous-frequency andits dual joint time and instantaneous-frequency transform toobtain a sparse representation of a signal that is not sparse intime or frequency, or sparse in either of these domains.

In [7], the discrete linear chirp transform (DLCT) is in-troduced to provide a linear-chirp representation of signals.The DCLT is a joint frequency and instantaneous frequencytransform. It is applicable in the analysis of non-stationarysignals, it can be implemented with the fast Fourier trans-form, and it provides a dual transform jointly relating timeand instantaneous-frequency. A clear application for theDLCT is compression of signals that are sparse in time orin frequency, or in neither time nor frequency. The DLCTparameters characterizing the signal would permit us to gen-erate a compressed form useful for transmission or storage.This paper will provide the necessary information about theDLCT and illustrate its application to compression. Testsignals will be used to compare our procedure with CS.

The rest of the paper is organized as follows. In section2 we discuss the DLCT transform, so that in section 3 and4 we are able to propose and demonstrate its application todata compression. Although our procedure is illustrated withtest signals that are not necessarily sparse in either time orfrequency, it is applicable to signals that are sparse in time orin frequency.

20th European Signal Processing Conference (EUSIPCO 2012) Bucharest, Romania, August 27 - 31, 2012

© EURASIP, 2012 - ISSN 2076-1465 2128

2. THE DISCRETE LINEAR CHIRP TRANSFORM(DLCT)

Sparseness or compressibility is fundamental in the transmis-sion and storage of signals. Sparseness can be obtained, insome cases, using the uncertainty principle. For instance, asinusoid of infinite time support is sparse in frequency, whilean impulse is sparse in time but not in frequency. As proposedin [4], for a given signal one can select either the time or thefrequency domain for which the signal is sparser, by forcingthe sparseness through thresholding the transform of the sig-nal. Using a random measurement matrix, the reconstructionof sparse signals is converted into a convex optimization. De-termining the sparseness of signals requires considering jointtime-frequency transformations.

A chirp signal would be an example of a signal that is notsparse in time or in frequency, although it could be consid-ered sparse in an intermediate domain. This points toward theapplication of the Fractional Fourier transform (FrFT) [8, 9]or of the polynomial time-frequency transforms [10]. Whatseems to be needed is a transformation that represents anysignal in terms of chirps, having modulation and duality prop-erties. Such a representation would determine if the signalis sparse, and if not how to transform it into an equivalentsparse signal. In [7] a local signal representation in terms oflinear chirps is proposed. The modulation and duality prop-erties of this transform permit us to change the given signalinto a sparser domain, and to consider the sinusoidal and theimpulse representations as special cases.

Given a discrete-time signal x(n), 0 n N � 1, itsdiscrete linear chirp transform (DLCT) is given as [7]

X(k,�) =N�1X

n=0

x(n) exp

✓�j 2⇡

N(�n2 + kn)

◆

0 n, k N � 1, � ⇤ � < ⇤. (1)

The DLCT is obtained from a basis of linear chirps

��,k

(n) = exp

✓j2⇡

N(�n2 + kn)

◆

characterized by a chirp rate �, a continuous variable con-nected with the instantaneous frequency of the chirp:

IF (n, k) =2⇡

N(2�n+ k),

and by the discrete frequency 2⇡k/N . Assuming a finite sup-port for �, i.e., �⇤ � < ⇤, it is possible to construct anorthonormal basis {�

�,k

(n)} with respect to k in the supportsof � and n. To obtain a discrete transformation, we approxi-mate the chirp rate as

� ⇡ `C, where C = 2⇤L

so that

�L2

` L2

� 1 integer.

The inverse discrete linear chirp transform is proposed in[7] as

x(n) =

L/2�1X

`=�L/2

N�1X

k=0

X(k,�)

LNexp

✓j2⇡

N(�n2 + kn)

◆

0 n, k N � 1, � ⇤ � < ⇤. (2)

The DLCT is a joint instantaneous-frequency frequencytransform that generalizes the discrete Fourier transform(DFT) as X(k, 0) is the DFT of x(n).

A dual transformation is obtained by interchanging thetime and frequency variables in (1) and 2 :

x̂(n,�) =N�1X

k=0

ˆX(k) exp

✓j2⇡

N(�k2 + nk)

◆

ˆX(k) =

L/2�1X

`=�L/2

N�1X

n=0

x̂(n,�)

LNexp

✓�j 2⇡

N(�k2 + nk)

◆

0 n, k N � 1, � ⇤ � < ⇤. (3)

It can be shown that ˆX(k) is the DFT of x(n) or X(k, 0).Thus, the DLCT can be used to represent signals that are com-binations of sinusoids or chirps of small chirp rates, while thedual DLCT is more appropriate for signals that are combina-tions of impulses or chirps of large chirp rates.

It is important to remark that in a discrete chirp, obtainedby sampling a continuous chirp satisfying the Nyquist criteria,the chirp rate � cannot be an integer. Indeed, if a finite supportcontinuous chirp

x(t) = ej(↵t2+⌦0t) ↵ =

�⌦

2�t, 0 t T

is sampled using ⌦s

= 2⇡/Ts

= M⌦max

, M � 2, the dis-crete signal is

x(n) = x(t)|t=nT

s

= ej([↵T2s

]n2+[⌦0Ts]n)

= ej(�̂n2+!

o

n)0 n T/Ts = N � 1

where we let ˆ� = ↵T 2s

be the chirp rate and !0 = ⌦0Ts bethe discrete frequency. Then the modulated chirp

x(n)e�j!0n = ej�̂n2

where

ˆ� = ↵T 2s

=

�⌦

2�t

�T 2s

=

⌦

s

Ts

/M

2N=

⇡/M

N

therefore,

� =N ˆ�

2⇡=

1

2Mis not an integer for M � 2. The discrete chirp-Fourier trans-form proposed in [12], assumes the chirp rate � is an integer— indicating that if the discrete chirp is obtained by sam-pling a continuous chirp it is aliased. For not aliased chirps,we need |�| 0.25.

2129

For each value of � it can be shown that

x�

(n) =N�1X

k=0

X(k,�)

Nexp

✓j2⇡

N(�n2 + kn)

◆

equals x(n) so that the inverse DLCT is the average over allvalues of �.

3. SIGNAL COMPRESSION USING THE DLCT

The main goal of signal compression is to reduce the amountof data that we want to transmit or store. The direct and thedual DLCT are used to represent signals that can be betterrepresented by one of them locally. Considering that a sinu-soid has a chirp rate � = 0, while an impulse has as chirprate � ! 1, we separate signals into two groups: one having0 |�| 0.5, corresponding to a linear chirp with a slopewith an angle in [�45o 45o], and the other for 0.5 < |�| < 1corresponding to a linear chirp with a slope with an angle in[45

o, 90o] or [�45o, �90o]. The value of � = 0.5 is not arbi-trarily chosen since it relates to the slope of the instantaneousfrequency such that

Slope = tan(✓) = 2�

If � = 0.5, then ✓ = ⇡/4 which is the angle that separates thetime-frequency space into two symmetric halves.

The performance of the proposed algorithm is measuredby signal to noise ratio (SNR) and the compression ratio (Cr),

SNR = 10 log✓�2x

�2e

◆

Cr =length of original signal

length of compressed signal

where �2x

is the mean square of the original signal and �2e

isthe mean square of the error signal or the difference betweenthe original and the reconstructed signals. Another factor thatplays an important role in compression is a threshold. Aftercalculating the DLCT of a signal, many of the coefficients ofthe resulted signal are close to or equal to zero. Thus, we canmodify those coefficients to produce more zeros by zeroingout them using certain threshold.

3.1. The proposed compression algorithm

In this section, we present a new algorithm for signal com-pression using DLCT. Figure 1 shows the block diagram ofthe proposed method.

Consider the local representation of a signal x(n), 0 n N � 1, as a superposition of P linear chirps

x(n) =P�1X

i=0

ai

exp

✓j2⇡

N(�

i

n2 + ki

n) + j'i

◆

= x{|�i

|0.5}(n) + x{|�i

|>0.5}(n)

where {ai

, 'i

, ki

, �i

} are the amplitude, phase, frequency,and chirp rate of the ith linear chirp. The algorithm has twopaths for the signal, the upper which is the dual path and thelower which is the direct path. Depending on the minimumvalue of the extracted �s for certain segment of the signal, wecan do the compression either by the dual path or by the directpath. The coefficients {a

i

, 'i

, ki

, �i

} are extracted and fromtheses coefficients we can reconstruct an approximation forthe signal x(n)— where the arrangement of these coefficientsis done according to the proposed data structure as will beshown in Fig. 2.

DLCT

FFT IDLCT Dual of DLCT

compress 𝛽 ≤ 0.5 𝑥(𝑛) to transmit

or store

No

Yes

Fig. 1. Compression algorithm.

3.2. The Developed Data Structure

The proposed data structure for sending or storing the ex-tracted parameters is shown in Fig. 2, we choose P chirprates that correspond to the peaks of chirps which forms thesignal and P is the order of the chirp model. Then, from eachvector which corresponds to the chosen chirp rates from thechirp transform X(k,�) or x̂(n,�) matrix, we select M

j

am-plitudes, phases, and frequencies or samples that have morepower of the signal concentrated upon them. .

Time

Amplitude

# of frequencies at each chirp rate

# of samples at each chirp rate

𝛽 ≤ 0.5 𝛽 > 0.5 𝑎 𝜑 𝑘 𝑎 𝜑 𝑛

Fig. 2. Data structure.

4. SIMULATION RESULTS

In this section, we present three experiment to illustrate theperformance of the proposed method, compare the resultswith the compressive sensing method, and explore the rela-tion between the chirp rate and the proposed algorithm.

Compressive sensing (CS) is a compression technique thatuses a fixed set of linear measurements providing that the sig-nal is sparse. The signal can be reconstructed by a convex

2130

optimization process. Consider the real and finite length sig-nal x[n] represented by its coefficient vector x 2 RN . Let usassume that the basis = [ 1... N ] where is an N ⇥ Nmatrix, the signal can be expressed in terms of the basis as [3]

x =

NX

i=1

si

i

or x = s

where s is a vector of size N ⇥ 1. The basis can be anyfunction that transforms x into a sparse signal. For instance,we can use sinusoidal basis such as discrete cosine transform.The signal s is a sparse signal in the new space and has Knonzero coefficients. Assume that the K nonzero coefficientsare not extracted directly, but we project the vector s onto amatrix � of size M⇥N where M < N . The matrix � is calledthe measurement matrix and it satisfies the condition that thecolumns of the sparsity basis cannot sparsely represent therows of the measurement matrix �. The measurements y canbe obtained as follows

y = � x = � s = ✓s

where y is a vector of size M ⇥ 1. The reconstruction of thesignal can be done via `1-optimization

ˆ

s = argmin ksk1 subject to y = ✓s

as shown in [4].In the first experiment, we use a segment of speech (1024

samples, sampling rate fs

= 8kHz) as shown in Fig. 3(a).Figure 3(b) and (c) give the magnitude of the DLCT and theWigner distribution for this segment of speech. The compres-sion ratio versus the SNR plot is shown in Fig. 3 (d). Sinceour goal is to obtain high SNR with high compression ratio,the proposed method gives more compression ratio than com-pressive sensing method, for an acceptable SNR. This seg-ment of speech has very small chirp rates at high frequencycomponents, with low concentrated energy, and sinusoids atlow frequency components with high concentrated energy.Since the minimum value � is less than 0.5, the compres-sion is obtained by the direct path. Even though, this segmentof speech can be considered a sparse signal in the frequencydomain, the proposed algorithm outperforms the compressivesensing method.

In the second experiment, a bird song signal (2048 sam-ples and sampling rate f

s

= 7, 350Hz) with � = 0.88 isconsidered; see Fig. 4 (a). This signal is sparser in the timedomain than in the frequency domain. Its dual DLCT and itsWigner distribution are shown in Figs. 4(b) and (c). Figure4(d) displays SNR versus compression ratio. In this experi-ment, the minimum value of � is greater than 0.5. Thus, thedual path is used for the compression. The proposed methodperforms better than CS method.

In the third experiment, we consider the case of a birdsignal (number of samples= 2048 and f

s

= 7, 350Hz) with

0 0.02 0.04 0.06 0.08 0.1 0.12−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (Sec)

Spee

ch si

gnal

(a)

k

β

0 50 100 150 200 250−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

(b)

0 0.02 0.04 0.06 0.08 0.1 0.12

246

x 10−3

Time (sec)

0

50

100

150

200

250

300

350

400

450

500

k

(c)

100100

101

SNR

(dB)

Compression ratio

Proposed methodCS method

(d)

Fig. 3. Experiment 1: (a) Segment of speech; (b) |X(k,�)|in two-dimensions; (c) The Wigner distribution of the signalshowing time and frequency marginals; (d) Compression ratiovs SNR for different methods.

� = 0.08 and � = 0.88. Figures 5 (a), (b), and (c) showthe signal, the magnitude of the DLCT, and the Wigner dis-tribution of the signal. Since, this signal is a chirp based, theimprovement of compression ratio for certain SNR is verylarge and it clearly shows the effectiveness of the proposedmethod over compressing sensing method as shown in Fig. 5(d). The minimum value in this experiment is � = 0.08, sothe compression is obtained by the direct path.

5. CONCLUSIONS

In this paper, we present a new algorithm for signal compres-sion based on the discrete linear-chirp transform (DLCT) andits dual. The extracted coefficients can be arranged in the de-veloped data structure. The simulation results show the effec-tiveness of the proposed method over the compressive sensing(CS) method. The improvement in the compression ratio de-pends on the nature of the signal. The effect of chirp rateon the performance of the direct and dual paths is also in-vestigated. It turns out the compression ratio depends on theminimum chirp rate of the linear chirps that forms the signal.The value of � = 0.5 is the decision maker. If �

mim

0.5we use direct path, otherwise dual path is used.

2131

0 0.05 0.1 0.15

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (Sec)

Bird

song

sign

al

(a)

nβ

0 200 400 600 800 1000 1200 1400 1600 1800 2000

−1.5

−1

−0.5

0

0.5

1

1.5

(b)

0 0.05 0.1 0.15 0.2 0.252468

x 10−3

Time (sec)

0

100

200

300

400

500

600

700

800

900

1000

k

(c)

100 101 102

101SN

R (d

B)

Compression ratio

Proposed methodCS method

(d)

Fig. 4. Experiment 2: (a) Bird chirping; (b) |X(k,�)| in two-dimensions; (c) The Wigner distribution of the signal showingtime and frequency marginals; (d) Compression ratio vs SNRfor different methods.

6. REFERENCES

[1] A. Gersho, “Advances in speech and audio compression,”Proc. of IEEE, vol. 82, no. 6, pp. 900-918, June 1994.

[2] J. D. Gibson, “Speech coding methods, standards, andapplications,” IEEE Circuits and Systems Magazine, vol.5, no. 4, pp. 30-49, Fourth Quarter 2005.

[3] R. G. Baraniuk, “Compressive sensing,” IEEE Sig. Pro-cessing Mag., vol. 24, no. 4, pp. 118-121, July 2007.

[4] E. J. Candes and J. Romberg, Toolbox-MAGIC,California Inst. Of Technology, Pasadena, CA,(http://www.acm.caltech.edu/l1magic/).

[5] E. J. Candes, “Compressive sampling,” Proc. of the In-ternational Congress of Mathematicians, Madrid, Spain,2006, pp. 1-20.

[6] G. Peyre, “Best basis compressed sensing,” IEEE Trans.on Sig. Processing, vol. 58, no. 5, pp. 2613-2622, May2010.

[7] O. A. Alkishriwo, and L. F. Chaparro, “A Discrete Lin-ear Chirp Transform (DLCT)for Data Compression,” in

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Two

over

lapp

ed b

ird si

gnal

s

(a)

k

β

0 200 400 600 800 1000 1200 1400 1600 1800 2000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

(b)

0 0.05 0.1 0.15 0.2 0.25

51015

x 10−3

Time (sec)

0

100

200

300

400

500

600

700

800

900

1000

k

(c)

100 101

101

SNR

(dB)

Compression ratio

Proposed methodCS method

(d)

Fig. 5. Experiment 3: (a) Bird chirping; (b) |X(k,�)| in two-dimensions; (c) The Wigner distribution of the signal showingtime and frequency marginals; (d) Compression ratio vs SNRfor different methods.

Proc. of the IEEE International Conference on Informa-tion Science, Signal Processing, and their Applications,submitted, Montreal, Canada, Jul. 2012.

[8] H. Ozaktas, Z. Zalevsky, and M. Kutay, The FractionalFourier Transform: with Applications in Optics and Sig-nal Processing, John Wiley & Sons, 2001.

[9] S. Senay, L. F. Chaparro, and L. Durak, “ Reconstructionof nonuniformly sampled time-limited signals using pro-late spheroidal wave functions,” Signal Processing, pp.2585-2595, 2009.

[10] A. Akansu, and H. Agirman-Tosun, “Generalized dis-crete Fourier transform with nonlinear phase,” IEEETrans. Sig. Processing, vol. 58, pp. 1-10, Sep. 2010.

[11] O. A. Alkishriwo, L. F. Chaparro, and A. Akan, “Sig-nal separation in the Wigner distribution using fractionalFourier transform,” European Signal Processing Conf.,EUSIPCO , Spain, pp. 1879-1883, Sep. 2011.

[12] X.-G. Xia, “Discrete chirp-Fourier transform and its ap-plication to chirp rate estimation,” IEEE Trans. on SignalProcessing, vol. 48, no. 11, pp. 3122-3133, Nov. 2000.

2132