ENERGY EFFICIENT OPTIMUM SAMPLING RATE FOR ANALOGUE SIGNALS WITH EXTREMELY WIDE BANDWIDTH USING COMPRESSIVE SENSING

International Journal of Wireless & Mobile Networks (IJWMN), Vol.14, No.3, June 2022
DOI:10.5121/ijwmn.2022.14305 73
ENERGY EFFICIENT OPTIMUM SAMPLING RATE
FOR ANALOGUE SIGNALS WITH EXTREMELY WIDE
BANDWIDTH USING COMPRESSIVE SENSING
Sylvester Ajah1
, Ifenyiwa, E. Achumba2
,
Nkwachukwu Chukwuchekwa3
, Nosiri Onyebuchi4
1,3,4
Department of Electrical /Electronic Engineering,
Federal University of Technology Owerri (FUTO), Imo State, Nigeria
2
Departement of Information and Communication Technology (ICT), Federal University of
Technology Owerri (FUTO), Imo State, Nigeria
ABSTRACT
The natural signals are mostly analogue in nature, but because of the benefits of digital processing of these
signals: flexibility, accuracy, storage and low cost; processing these signals digitally is often preferred.
But the existing analogues to digital converters are efficient in processing signals with small to medium
bandwidths, but inefficient for signals with large bandwidths. The real-time processing of these signals
with large bandwidths are done analogically or optically at the cost of the aforementioned advantages of
digital processing of these signals. This paper is aimed at solving the real-time challenge of processing
these extremely wide bandwidth signals digitally using a compressive sensing (CS) algorithm, with specific
detail on the ways the application of CS will enhance the energy efficiency of wireless communication
devices. Consequently, determine the throughput at which the use of CS is energy efficient for wireless
devices using energy-efficient compressive sensing throughput (EECST) model. The simulation results
show that the throughput requirements for introducing CS in any machine to machine (M2M) / internet of
things (IoT) communication application to be energy efficient are minimum of 54bits per second and 317
bits per second when the required number of clock cycles for performing various device applications is
20,000 and 50000 respectively.
KEYWORDS
Analog Signals, Analogue to Digital Converters, Compressive Sensing, Digital Signals, Digital Signal
Processing, Energy Efficiency, Extremely Wide Bandwidth Signals, Optimum Sampling Rate.
1. INTRODUCTION
Most signals in nature are analogue signals, to process these signals digitally, they have to be
converted to digital form using analogue – to – digital (A/D) converter. This conversion is done
via sampling, quantization, and coding the quantized sample into digital numbers. The conversion
of analogue signals to digital form result in distortion that inhibits the reconstruction of the
original analogue signals fully. The control of this distortion can be achieved by proper choice of
the sampling rate, and the precision in the quantization process. Sampling is the conversion of
continuous – time signal into a discrete - time signal gotten via taking samples of the continuous
time signals at discrete – time instants [1]. Figure 1 shows A/D conversion process.
However, many communication systems involve high bandwidth, but sparse radio frequency
(RF) signals. These features necessitate the application of compressive sensing technique in
enhancing the energy efficiency [2]. Compressive sensing (CS) is a signal processing strategy for

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efficient signal acquisitions and reconstructions, which sample the signals at approximately
information rates. It can also be described as signal acquisition method that collects few samples
of signal of interest and use optimization technique to reconstruct the original signal from
incomplete measurement. This technique has provided the window of opportunities for enhancing
the energy efficiency of machine to machine (M2M) / IoT communication systems.
CS combines the signal sampling and compression into a single process, with low sensitivity to
packet loss and graceful degradation of signal in the event of unusual sensor readings [3]. It also
reduces the time spent on data acquisition by the nodes via intelligently picking the coefficients
of the non-zero part of signals to be sensed. Hence reducing the energy cost of sensing and
communications. Also, CS represents a paradigm shift in which the number of measurements is
reduced during acquisition so that no additional compression will be needed [4]. Furthermore,
CS exploits the information rate with any signal, hence removes redundancy in the signal during
sampling process. Leading to lower effective sampling rate which reduces the energy cost of
sampling in the nodes. Also, most computations take place in the base stations (sink) in CS,
hence elongating the life span of M2M devices [3]. Note that many of these M2M / IoT
communication devices that are deployed or will be deployed will be only depend on their
internal battery capacities for processing their data, while their servers (receivers) in the data
centres will have access to power sources, hence the durability of the M2M / IoT communication
networks will mainly depend on the durability of the deployed M2M / IoT communication
devices that are batteries powered. This energy efficient communication paradigm has been
applied in signals processing, statistics, optimization and many other applications including
wireless communications [5]. Consequent to the aforementioned points, the imperativeness
energy efficient algorithms to elongate the battery life span of these devices.
This technique has provided the window of opportunities for enhancing the energy efficiency of
IoT communication systems, and optimum sampling of radio frequency signals with extremely
high bandwidth. It also reduces the time spent on data acquisition by the wireless devices via
intelligently picking the coefficients of the non-zero part of signals (frequency bandwidth) to be
sensed, hence reducing the energy cost of sensing and communications simultaneously. Also,
most computations take place in the base stations (sink) in CS algorithm, hence elongating the
life span of M2M communication devices [3]. These enumerated points make CS an effective
technique for enhancing the energy efficiency of wireless devices, and optimum sampling of the
extremely bandwidth analogue signals.
Figure 1. Analogue - to – digital conversion Process

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In figure 1, the analogue signal is represented by 𝑥𝑎(𝑡), when the signal is passed via sampler, it
resulted to discrete time signal represented by 𝑥𝑎(𝑛𝑇) ≡ 𝑥(𝑛), where T is the sampling
interval. The faster the sampling rates, the lower the sampling periods, the lower the distortion of
the analogue signal during reconstruction. But this result to more expensive A/D converter based
on the existing technology [1]. Hence this paper hope to address this challenge using a more
energy efficient paradigm in analogue to digital conversion process that is based on the existing
technology called compressive sensing. The discussion on this paper is limited to periodic
sampling as described in equation (1).
𝑥(𝑛) = 𝑥𝑎(𝑛𝑇) − ∞ < 𝑛 < ∞ (1)
The reciprocal of the sampling period T is called the sampling rate and it is represented in
equation (2).
1
𝑇
= 𝐹𝑠 (2)
The unit of sampling rate is samples per second or the sampling frequency in Hertz. The
relationship between the time variable (t) of the periodic analogue signal to (n) of the discrete
time signal is shown in equation (3).
𝑡 = 𝑛𝑇 =
𝑛
𝐹𝑠
(3)
From equation (3), there is a relationship between the frequency in the analogue signal F, and the
frequency variable in the discrete – time signals f. To establish this relationship, consider
analogue sinusoidal signal of the form shown in equation (4).
𝑥𝑎(𝑡) = 𝐴𝑐𝑜𝑠(2𝜋𝐹𝑡 + 𝜃) (4)
Where A is the amplitude of the sinusoidal signal, and 𝜃 is the phase angle. Which when they
are sampled periodically at a rate 𝐹𝑠, yields equation (5).
𝑥𝑎(𝑛𝑇) ≡ 𝑥(𝑛) = 𝐴𝑐𝑜𝑠(2𝜋𝐹𝑛𝑇 + 𝜃) = 𝐴𝑐𝑜𝑠 (
2𝜋𝑛𝐹
𝐹𝑠
+ 𝜃) (5)
A discrete time sinusoidal signal can be expressed in equation (6).
𝑥(𝑛) = 𝐴𝑐𝑜𝑠 (𝜔𝑛 + 𝜃) , −∞ < 𝑛 < ∞ (6)
Where 𝜔 ≡ 2𝜋𝑓
Therefore,
Therefore,
𝑥(𝑛) = 𝐴𝑐𝑜𝑠(2𝜋𝑓𝑛 + 𝜃), 𝑓𝑜𝑟 − ∞ < 𝑛 < ∞ (7)
Comparing equations (5) and (7) yields equation (8). It can be observed that the frequency
variable in the analogue signal F, and that of the digital signal f are linearly related as given in
(8).

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𝑓 =
𝐹
𝐹𝑠
≡ 𝜔 = Ω𝑇 (8)
Where f is the frequency of the digital signal, F is the frequency of the analogue signal and 𝐹𝑠is
the sampling frequency. Equation (8) is called relative or normalized frequency. Equation (8)
implied that the frequency of the digital frequency can be established from the frequency of the
analogue frequency if the 𝐹𝑠 is known.
If the analogue signal is converted to digital signal via sampling the analogue signal at discrete
time instants, obtaining a discrete – time signals as explained above, the process of converting
this discrete continuous time signal into discrete values is called quantization, which is
approximating the discrete time signals to a finite value. In equation (7), the variation of
continuous – time sinusoidal signals varies from −∞ < 𝑛 < ∞ but discrete – time sinusoidal
signals, the variation is as given in equation (9).
−
1
2
< 𝑓 <
1
2
≡ −𝜋 < 𝜔 < 𝜋 (9)
The fundamental difference between continuous – time signals and discrete – time signals is the
range values of the frequency variables F and f. Periodic sampling of continuous – time signals
implies the mapping of the infinite frequency range for the variable F into a finite frequency
range variable f. This is because the highest frequency in a discrete – time signal 𝜔 =
𝜋 𝑜𝑟 𝑓 =
1
2
. This follows that, with 𝐹𝑠 the corresponding highest values of F and Ω are
given in equation (10) [1].
𝐹𝑚𝑎𝑥 =
𝐹𝑠
2
=
1
2𝑇
𝑎𝑛𝑑 Ω𝑚𝑎𝑥 = 𝜋𝐹𝑠 =
𝜋
𝑇
(10)
The paper is organized as fellows; section 2 discussed the basis of sampling theorem, section 3
discussed sampling theorem, section 4 discussed the theory of comprehensive sensing, section 5
discussed energy consumption in wireless devices, section 6 discussed Energy Efficient
Compressed Sensing Throughput (EECST) Model, and section 7 concludes the work.
2. BASIS OF SAMPLING THEOREM
Given any analogue signal, how to select the sampling rate is core to be able to effectively
reconstruct with minimal distortion. Assume that any analogue signal can be represented as the
sum of sinusoids of different amplitudes, frequencies and phases as given in equation (11).
𝑥𝑎(𝑡) = ∑ 𝐴𝑖
𝑁
𝑖
cos(2𝜋𝐹𝑖𝑡 + 𝜃𝑖) (11)
Where N is the number of the frequency components. All the signals lend themselves to
representation in equation (11) over short time segment. The amplitude, frequencies and phases
usually change slowly with time from one-time segment to another. Moreover, assuming that the
frequencies do not exceed some known frequencies like 𝐹𝑚𝑎𝑥. Example, the 𝐹𝑚𝑎𝑥 for some
speech signal is 3000Hz, while the 𝐹𝑚𝑎𝑥 for television signal is 5000Hz for television signals.
Since the maximum frequency may vary slightly from different realization among signals of any
given class, they may be wish to ensure that 𝐹𝑚𝑎𝑥 does not exceed some predetermined value by

77
passing the analogue signal through a filter that severely attenuates the frequency components
above 𝐹𝑚𝑎𝑥 [1].
From the knowledge of 𝐹𝑚𝑎𝑥, the appropriate sampling rate can be selected. From Nyquist
theorem, the highest frequency in an analogue signal can be reconstructed when the signal is
sampled at a rate 𝐹𝑆 =
1
𝑇
𝑖𝑠
𝐹𝑠
2
. Any frequency above
𝐹𝑠
2
or below −
𝐹𝑠
2
results to a sample
that are identical with corresponding frequency in the range −
𝐹𝑠
2
≤ 𝐹 ≤
𝐹𝑠
2
. To avoid ambiguity
resulting from aliasing, the sampling rate that is sufficiently high must be selected. That is,
𝐹𝑠
2
must be greater than 𝐹𝑚𝑎𝑥. Thus to avoid the problem of aliasing, 𝐹𝑠 is selected so that equation
(12) is observed;
𝐹𝑠 > 2𝐹𝑚𝑎𝑥 (12)
As mentioned earlier, 𝐹𝑚𝑎𝑥is the highest frequency component in the analogue signal. With the
sampling rate selected in this manner, any frequency component say |𝐹𝑖| < 𝐹𝑚𝑎𝑥in the analogue
signal is mapped into a discrete – time sinusoid with a frequency as given in equation (13).
−
1
2
≤ 𝑓𝑖 =
𝑓𝑖
𝐹𝑠
≤
1
2
≡ −𝜋 ≤ 𝜔𝑖 = 2𝜋𝑓𝑖 ≤ 𝜋 (13)
Since |𝑓| =
1
2
𝑜𝑟 |𝜔| = 𝜋 is the highest (unique) frequency in a discrete – time signal, the choice
of sampling rate according to equation (12) avoids the problem of aliasing [1]. Alternatively, the
condition 𝐹𝑠 > 2𝐹𝑚𝑎𝑥ensure that all the sinusoidal components in the analogue signal are mapped
into discrete – time frequency components with frequencies in the fundamental interval. Thus
analogue signals can be reconstructed without distortion from the sample values using
appropriate interpolation formula.
3. SAMPLING THEOREM
If the highest frequency contained in an analogue signal 𝑥𝑎(𝑡) is 𝐹𝑚𝑎𝑥 = 𝐵 , and the signal has
a rate 𝐹𝑠 > 2𝐹𝑚𝑎𝑥 ≡ 2𝐵 , then 𝑥𝑎(𝑡) can exactly be recovered from its sample values using
interpolation function as given in equation (14).
𝑔(𝑡) =
𝑠𝑖𝑛2𝜋𝐵𝑡
2𝜋𝐵𝑡
(14)
Hence, 𝑥𝑎(𝑡) may be expressed as given in equation (15).
𝑥𝑎(𝑡) = ∑ 𝑥𝑎 (
𝑛
𝐹𝑠
) 𝑔 (𝑡 −
𝑛
𝐹𝑠
)
∞
𝑛=−∞
(15)
Where 𝑥𝑎 (
𝑛
𝐹𝑠
) = 𝑥𝑎(𝑛𝑇) ≡ 𝑥(𝑛)is the sample of 𝑥𝑎(𝑡)
When the sampling of 𝑥𝑎(𝑡) is performed at the minimum sample rate 𝐹𝑠 = 2𝐵 , the
reconstruction formula in equation (15) becomes as given in equation (16).

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𝑥𝑎(𝑡) = ∑ 𝑥𝑎 (
𝑛
2𝐵
)
sin 2𝜋𝐵 (𝑡 −
𝑛
2𝐵
)
2𝜋𝐵 (𝑡 −
𝑛
2𝐵
)
∞
𝑛=−∞
(16)
The sample rate 𝐹𝑁 = 2𝐵 = 2𝐹𝑚𝑎𝑥 is called Nyquist rate and the above theorem is called
Nyquist sampling theorem, which is basically the theorem used in most digital signals processing.
4. THEORY OF COMPRESSIVE SENSING (CS)
The basic principle of CS is to transform code the analogue signal of interest x into the basis or
frame that will provide the compressible and sparse representation of the signal [6]. Transform
coding is done using Fourier transform (FT), fast Fourier transforms (FFT), discrete cosine
transforms (DCT), wavelet transforms (WT), etc. The sparse representation of the signal of
length n, entails that it can be represented with k nonzero coefficients, where 𝑘 ≪ 𝑛. While the
compressible representation of the signal entails that the signal can be well-approximated to those
k none zeros part of the signal n. The CS algorithm can represent the signal of interest with high
fidelity by preserving the k none zeros value of n and their locations. This method is called sparse
approximation, which is the foundation of transform coding schemes, that used the signal sparsity
and compressibility like JPEG, MPEG, MP3 and JPEG2000 standards [6]. The number of non-
zeros coefficient of x is less than or equal to k. With this technique, the extremely wide
bandwidth signals can be reduced to size in which the existing A/D can effectively and efficiently
sample the signals.
||𝒙||𝟎 ≤ 𝒌 (17)
The signal x is encoded into a smaller vector say b with the aid of a sensing matrix 𝑨 ∈ 𝑹𝒎 × 𝒏
,
where 𝑚 < 𝑛 and it is chosen independently of x [5]. The CS coded signal can be represented as
given in (17). The CS approach involves directly acquiring the compressed samples without
going through the intermediate stages, the compressive measurement through linear projections
as given in (18) [7].
𝑨𝒙 = 𝒃 (18)
In the applications of CS in M2M / IoT communication, the encoding of x is not calculated by a
computer or microcontroller, but obtained by certain electrical or electromagnetic, physical, or
optical measuring means, depending on the application. Also, because of the condition in
equation (19), b is the compression of x.
{
𝒌 ≤ 𝒎
𝒎 < 𝑛
(19)
Furthermore, on the application of this technique on M2M communication paradigm, b is
recorded by the wireless M2M device and becomes digitally available to the decoder. Though
equation (18) above is an underdetermined equation system and has infinite number of solutions,
x is recovered from b by finding the sparsest solution of equation (18) by solving equation (20).
||𝒙||𝟎
𝒙
𝒎𝒊𝒏
𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝑨 = 𝒃 (20)
Equation (20) above is called 𝑙0 norm, and though the combinational equation of (20) is Non-
deterministic Polynomial-time hard (NP Hard), and the method of trying all the possible supports

79
of cardinality k is computationally intractable. To make it tractable, 𝒍𝟎 norm is replaced by 𝒍𝟏
norm as given in equation (21).
||𝒙||𝟏
𝒙
𝒎𝒊𝒏
𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐 𝑨𝒙𝒃 (21)
Equation (21) above is a convex program and has several fast solvers than equation (20). In ideal
case, it is better to recover x from (21) when m equals 2k, x is uniquely determined by k indices,
and k values of its non-zeros entries. In summary, the whole processes of compressive sensing
consist of the following stages; signal sparse representation using the transform coding
techniques discussed above, linear encoding measurement collection using sensing matrix
discussed in the subsection that follows, and compressed signal recovery also discussed in
subsection 4.3. For linear encoding measurement collection, the types sensing matrix A as it is in
equation (18) is invaluable in determining the error free compressing and recoverability of the
signals b. The types of sensing matrices is as discussed in the subsection 4.2.
4.1. Sensing Matrices
The A in equation (18) denotes sensing matrix, which is 𝑀 × 𝑁 matrix, and represents a
dimensionality reduction because it maps 𝑅𝑁
to 𝑅𝑀
, where 𝑀 ≪ 𝑁. In designing the sensing
matrix, they are two fundamental theoretical questions that need to be addressed; how to design
the sensing matrix A to ensure that it preserves the information in the signal x, and how to recover
the original signal x from the compressed measurement b. To effectively address the above
fundamental questions, they are features A is expected to have to effectively address the
aforementioned challenges. These features are discussed below.
4.1.1. Null Space
The NSP of A which is given in equation (22) is very important in the recovery of the
compressed signal x’ from b using l1-minimization. The null space of all 𝐴 vectors x, which are
mapped to 0 is given in equation (22).
ℵ(𝐴) = {𝑥: 𝐴𝑥 = 0} (22)
This entails that two different k-sparse signals 𝑥 ≠ 𝑥′do not result to same measurement vector b.
This means that their difference is not part of the null space of the measuring matrix as given in
equation (23).
𝐴(𝑥 − 𝑥′
≠ 0 ↔ 𝑥 − 𝑥′
∉ ℵ(𝐴) (23)
For instance, the difference between two k-sparse vectors is at most 2k-sparse. Hence a k-sparse
x is uniquely defined if ℵ(𝐴) contains no 2k-sparse vectors. This entails that any 2k columns of A
are linearly independent, which results to lower bound of the number of measurements, that is the
necessary condition that reconstruction is possible, and is as given in equation (24) [8].
𝑀 ≥ 2𝑘 (24)
This is because to be able to recover all the sparse signals x from the measurement b, then it is
imperative that 𝐴𝑥 ≠ 𝐴𝑥′ so that it will be possible to distinguish x from x’ based on b [9]. The
sensing matrix 𝐴 ∈ 𝑅𝑀×𝑁
is said to satisfy the null space property (NSP) of order k with
constant 𝛾 ∈ (0,1) if the condition in equation (25) is met by A.

80
||𝜂Τ||1 ≤ 𝛾||𝜂𝑇𝑐||1 (25)
Where 𝑇 ⊂ {1, … … … … … , 𝑁} 𝑎𝑛𝑑 𝑇 ≤ 𝑘, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜂 ∈ ker 𝐴 ≡ set of the k-largest entries of x
in the absolute value [10]. Though, there are many ways of characterizing this property of 𝐴 , the
most used property in characterizing the NS property is the spark ( 𝜎 = 𝑠𝑝𝑎𝑟𝑘(𝐴) ) of the
sensing matrix A. The spark of A is the minimum numbers of columns of A that are linearly
dependent [11]. For instance, when there are non-zero columns in A, then 𝜎 ≥ 2, with equality
only when two columns in A are linearly dependent [12].
4.1.2. Restricted Isometry Property (RIP)
This is the property of the sensing matrix that aids it to recover the compressed signal that is
contaminated with noise. A sensing matrix A is said to satisfy the RIP of order K if there exist a
restricted isometry constant 𝜕𝐾 ∈ (0, 1) of a matrix 𝐴 ∈ 𝑅𝑀×𝑁
that is the smallest number, such
that the condition in equation (26) holds for all 𝑥 ∈ ∑ =
𝐾 {𝑥:||𝑥||0 ≤ 𝐾. RIP is also called
uniform uncertainty principle[13].
(1 − 𝜕𝐾) ||𝑥||2
2
≤ ||𝐴𝑥||2
2
≤ (1 + 𝜕𝐾)|| 𝑥||2
2
(26)
If the matrix A satisfies the RIP of the order 2K, then equation (26) can be explained that A
approximately preserve the distance between any pair of K-sparse vectors. In the definition of the
RIP above, bounds that are symmetric about 1 was assumed. In practice, arbitrary bounds as
given in equation (27) could be used.
𝛼 ||𝑥||2
2
≤ ||𝐴𝑥||2
2
≤ 𝛽|| 𝑥||2
2
(27)
4.1.3. Incoherence
𝜇(𝐴) =
|< 𝐴𝑖, 𝐴𝑗 >|
||𝐴𝑖||2||𝐴𝑗||2
1≤𝑖≤𝑗≤𝑁
𝑚𝑎𝑥
(28)
It has been established that 𝐴 with coherence coefficient 𝜇 satisfies the RIP of 𝑠 with
𝛿𝑠 ≤ 𝜇(𝑠 − 1) whenever 𝑠 <
1
𝜇
+ 1 , where s is the sparcity level of the sparse signal [15]
[16].

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𝜇 ≥ √
𝑁 − 𝑀
𝑀(𝑁 − 1)
(29)
𝑘 <
1
2
(1 +
1
𝜇(𝐴)
) (30)
4.2. Types of Sensing Matrices
Compressive sensing involves three main steps: sparse representation, measurement and sparse
recovery. The sensing or measurement matrix are used to sample only the components that best
represent the sparse signal. Hence, the choice of the sensing matrix affects the success of the
sparse recovery process. Therefore, the design of an accurate sensing matrix is a vital process in
compressive sensing [19]. The authors in [20] stated that design of measuring matrix and
development of an efficient sparse recovery algorithms are the two main problems that must be
addressed by compressive sensing theory.
Sensing matrices are broadly classified into two groups – random sensing matrices and
deterministic sensing matrices. Random sensing matrices are generated at random by identical or
independent distributions (iid) like normal, Bernoulli random and Fourier ensembles, they are
easy to construct and they satisfy the RIP with high probability. Random sensing matrices are
further classified into two types – structured and unstructured sensing matrices as shown in figure
2. While the deterministic sensing matrices are constructed deterministically to satisfy the RIP, or
to have small mutual coherence. Most of the deterministic sensing matrices are made based on
employing polynomials of finite fields [16].

82
Figure 2. Classification of Sensing Matrices [19]
4.3. Compressed Signals Recovery
The recovery of the compressed signal sample is the critical part in determining its adoption in
M2M communications paradigm. This sub-section provides the summary of the algorithms that
are used for sparse compressed signals recovery. The compressed signal recovery is NP – hard
(Non deterministic polynomial-time hard), hence can be recovery using alternative algorithms.
These recovery algorithms can broadly be classified into three; the convex and relaxation
algorithms, the Bayesian algorithms and Greedy algorithms. The convex and relaxation
algorithms solve the sparse signals recovery via convex relaxation algorithms; the examples are
as shown in figure 3. The Greedy algorithms solve the sparse signals recovery via iterative
processes, examples of this type of algorithm are also shown in figure 3. Bayesian algorithms
solve the sparse signals recovery via taking into account a prior knowledge of the sparse
distributions, examples are also shown in figure 3.
5. ENERGY CONSUMPTION IN WIRELESS M2M COMMUNICATIONS
DEVICES
Equation (31) summarizes various energy costs associated with wireless communication, and
figure 4 gives the percentage of various operational energy costs within the wireless devices.
These energy costs are based on MAC protocol, because it has a major impact on the power
consumption of the wireless devices, and it is based on a single sampling period [4].
𝑬𝑇 = 𝑬𝑠𝑡𝑎𝑟𝑡−𝑢𝑝 + 𝑬𝑟𝑎𝑚𝑝 + 𝑬𝑠𝑒𝑛𝑠𝑖𝑛𝑔 + 𝑬𝑙𝑜𝑔𝑔𝑖𝑛𝑔 + 𝑬𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 + 𝐸𝑐𝑜𝑚𝑝𝑢𝑡𝑖𝑛𝑔
+ 𝑬𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑜𝑛 (31)
Where 𝑬𝑻 is the total operational energy consumption costs in the wireless devices; 𝑬𝒔𝒕𝒂𝒓𝒕−𝒖𝒑 is
the device initial energy consumption during start-up; 𝑬𝒓𝒂𝒎𝒑 is the energy cost of switching to

83
different energy states for various devices’ operations; 𝐸𝑠𝑒𝑛𝑠𝑖𝑛𝑔 is the energy used by the wireless
devices for sensing signals; 𝑬𝒍𝒐𝒈𝒈𝒊𝒏𝒈 is the energy cost of storing the sensed data; 𝑬𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈 is
the data sampling energy cost, 𝑬𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 is the energy used by the wireless devices in internal
processing of the sensed signals; and 𝑬𝒄𝒐𝒎𝒎𝒖𝒏𝒊𝒄𝒂𝒕𝒊𝒐𝒏 is the communication energy cost used for
transmitting digital signals to the receiver.
Being that start-up energy cost in the wireless device is negligible (< 1%) as can be seen in figure
4, its effect will not be considered further. The ramp energy cost in the micro-controller unit
(MCU) of the wireless devices is negligible too, but the ramp energy cost in the radio is
significant and it constitutes 10% of the total energy cost in the wireless device. Equation (32) is
used for calculating the radio ramp energy cost.
Figure 3. Classifications and examples of sparse recovery algorithms
Figure 4. Various operational energy costs in the wireless device
𝑬𝒓𝒂𝒎𝒑 =
|(𝑰𝒔𝒕𝟐 − 𝑰𝒔𝒕𝟏)| × 𝑻𝒔𝒕𝟏𝟐 × 𝑽𝒅𝒄
𝟐
(32)
Where 𝑰𝒔𝒕𝟐 is the current consumed in the state switched to, 𝑰𝒔𝒕𝟏 is the consumed current in the
current state, 𝑻𝒔𝒕𝟏𝟐 is time used in switching from state 1 to state 2, and 𝑽𝒅𝒄 is the voltage
consumed.
The sensing energy cost is about 6% of the total energy cost in the wireless device as can be seen
in figure 2, and the energy cost of sensing b bits of data can be calculated using equation (33).

84
𝑬𝒔𝒆𝒏𝒔𝒊𝒏𝒈 = 𝒃 × 𝑽𝒅𝒄 × 𝑰𝑺𝒆𝒏𝒔𝒊𝒏𝒈 × 𝑻𝒔𝒆𝒏𝒔𝒊𝒏𝒈 (33)
Where 𝑰𝑺𝒆𝒏𝒔𝒊𝒏𝒈 and 𝑻𝒔𝒆𝒏𝒔𝒊𝒏𝒈 are the current consumed in sensing and the sensing time
respectively.
The logging energy cost is the energy used by the wireless device for reading b bit packet data
and writing it into the memory [21]. Equation (34) shows how to evaluate the energy cost of
logging bbits’ data size per cycle.
𝑬𝒍𝒐𝒈𝒈𝒊𝒏𝒈(𝒃) = 𝑬𝒓𝒆𝒂𝒅 + 𝑬𝒘𝒓𝒊𝒕𝒆 =
𝒃 × 𝑽𝒅𝒄
𝟖
(𝑰𝒓𝒆𝒂𝒅 × 𝑻𝒓𝒆𝒂𝒅 + 𝑰𝒘𝒓𝒊𝒕𝒆 × 𝑻𝒘𝒓𝒊𝒕𝒆) (34)
The sampling energy cost 𝑬𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈 is the energy the wireless device spent on sampling b bits of
data. The 𝑬𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈 is greatly dependent on the type of data and the rate of sampling. Let the
period for data sampling be T, time spent on sampling data be 𝑡𝑑𝑎𝑡𝑎 , current used for data
sampling be 𝐼𝑑𝑎𝑡𝑎 and voltage used to be 𝑉𝑑𝑐. The energy spent on data sampling 𝑬𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈
is given in (35).
𝒕𝒃 = 𝒕𝒅𝒂𝒕𝒂 × (
𝟏
𝑻
) 𝑬𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈 = 𝒕𝒃 × 𝑰𝒅𝒂𝒕𝒂 × 𝑽𝒅𝒄 (35)
The energy cost of computing 𝑬𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 in a wireless communication device is a key
constituent of total energy cost in a wireless device. The 𝑬𝑪𝒐𝒎𝒑𝒖𝒕𝒊𝒏𝒈 consists of MCU’s active
and other modes. The computation energy 𝑬𝒄𝒐𝒎𝒑𝒖𝒕𝒂𝒕𝒊𝒐𝒏 is given in (36) in two states (active and
sleep).
𝑬𝒄𝒐𝒎𝒑𝒖𝒕𝒂𝒕𝒊𝒐𝒏 = 𝑽𝒅𝒄 × 𝑰𝒂𝒄𝒕𝒊𝒗𝒆 × 𝑻𝒂𝒄𝒕𝒊𝒗𝒆 + 𝑽𝒅𝒄 × 𝑰𝒔𝒍𝒆𝒆𝒑 × 𝑻𝒔𝒍𝒆𝒆𝒑 (36)
The communications energy costs consist of the energy cost of data transmission and the cost of
data reception is given in equation (37).
𝑬𝒄𝒐𝒎𝒎𝒖𝒏𝒊𝒄𝒂𝒕𝒊𝒐𝒏 = 𝑬𝒕𝒙 + 𝑬𝒓𝒙 (37)
Where 𝑬𝒕𝒙 is the energy cost of transmission of packets of data, and 𝑬𝒓𝒙 is the energy cost of
receiving data packets. Equations (38) and (39) represents 𝑬𝒕𝒙 and 𝑬𝒓𝒙 respectively.
𝑬𝒕𝒙 = 𝑽𝒅𝒄 × 𝑰𝒕𝒙 × 𝑩𝒍𝒕𝒙 × 𝑻𝒃𝒕𝒙 (38)
𝑬𝒓𝒙 = 𝑽𝒅𝒄 × 𝑰𝒓𝒙 × 𝑩𝒍𝒓𝒙 × 𝑻𝒃𝒓𝒙 (39)
Where 𝑰𝒓𝒙 𝑎𝑛𝑑 𝑰𝒕𝒙 are the current consumed in the reception and transmission mode
respectively; 𝑩𝒍𝒕𝒙 𝑎𝑛𝑑 𝑩𝒍𝒓𝒙 are the bit length of the transmitted, and received packets along with
their preambles respectively; 𝑻𝒃𝒕𝒙 𝑎𝑛𝑑 𝑻𝒃𝒓𝒙 are the time for transmitting and receiving single bit
of data. However in wireless communication devices, the energy costs of various operations are
evaluated in terms of the number of clock cycles required to perform such operations [22]. To
this end, the energy cost of performing various operations in wireless communication devices as
given in equation (31) are evaluated in terms of the number of clock cycles required to perform
such operations, which varies from one operation to another.

85
6. ENERGY EFFICIENT COMPRESSED SENSING THROUGHPUT (EECST)
MODEL
Equation (40) gives the fundamental communication model through which any communication
system follows, hence to have an energy efficient system, the effect of the energy cost of
variables in equation (40) have to be considered.
𝒚 = 𝑯𝒃 + 𝒏 (31)
Where y is the signal at the receiver, H is the channel properties, b is the compressed transmitted
signal as given in equation (18), and n is the channel noise. In (40), H (the channel
characteristics) has the most significant effect on the received signal y. The sensitivity of the
receiver is the major determining factor in knowing the amount of power required to get a signal
b from transmitting station to the receiver when the H properties have been ascertained using the
appropriate models. The receiver’s sensitivity is the minimum input signal (𝑆𝑚𝑖𝑛) required to
produce a specific output signal with a specified signal-to-noise ratio (S/N) and is given in
equation (41) [23].
𝑺𝒎 = (𝑺/𝑵)𝒎𝒊𝒏 × 𝑲 × 𝑻𝒐 × 𝑩 × (𝑵𝑭) (41)
Where (𝑺/𝑵)𝒎𝒊𝒏 is the minimum signal-to-noise ratio needed to detect a signal, NF is the noise
factor, K is Boltzmann’s constant =1.38 × 10−23
𝐽𝑜𝑢𝑙𝑒/°𝐾, 𝑇𝑜 is the absolute temperature of the
receiver input ( °𝐾𝑒𝑙𝑣𝑖𝑛) = 290°𝐾 and B is the receiver bandwidth (Hz) [24]. A typical
receiver’s sensitivity is around -110dBm, though it is dependent on device type [25].
As stated in [26], that sub 1GHz spectrum provides the energy efficient medium through which
M2M devices can be connected, let the channel frequency be Sub 1GHz Industrial, Scientific and
Medical (ISM) radio band (902 – 928) MHz with the centre frequency of 915MHz and bandwidth
of 26MHz. Now, to determine the power required to transmit the signal from the transmitter to
the receiver say 4 kilometres in a large city scenario, the path loss normally follows Rayleigh
distribution. For this discussion, let assume that the value of n as contained in (40) is zero. Also,
let the base station (BS) height be 40 meters, and mobile station height be 2 meters. Using Hata
path loss model [27] in path loss evaluation, using the above variables will result to a path loss of
144.55dB.
However, the channel losses between a transmitter, and a receiver in wireless channel is given in
equation (42).
𝐿𝑜𝑠𝑠𝑒𝑠 = 𝑃𝑎𝑡ℎ 𝑙𝑜𝑠𝑠 + 𝑝𝑒𝑛𝑒𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑙𝑜𝑠𝑠 + 𝑜𝑡ℎ𝑒𝑟 𝑙𝑜𝑠𝑠𝑒𝑠 (42)
The penetration loss can be evaluated using equation (43).
𝑳𝒑𝒍(∅) = √𝒅 − 𝒆(∅ − 𝒇)𝟐 (43)
Where d, e, f are empirical parameters; and ∅ is the angle of the signal inclination [28]. Now
using the experimental results for the empirical parameters as contained in [28] for computation.
The penetration loss at 60° signal reception angle is 23.69dB. Also, other losses as contained in
(33) are gotten by calculating 10% of the summation of the above losses. Therefore, other loss is
16.82dB. Hence, a Total loss as given (42) is approximately 185dB.
The power of the received signal at the receiver 𝑃𝑟 is given in (44).

86
𝑷𝒓 = 𝑷𝒕 + 𝑮𝒕 − 𝑳𝒐𝒔𝒔𝒆𝒔 (44)
Where 𝑃𝑡 is the transmit power at the transmitter and 𝐺𝑡 is the total gains in both the transmitter
and the receiver. Now using 𝑃𝑟 as -110dBm, assuming that the total gains between the transmitter
and receiver at 2dBm each is 4dB, then the required 𝑃𝑡 is 71dBm. This value of 𝑃𝑡 is the
minimum required to get any signal across 4 kilometres, assuming all the variables are constant.
Given that there are numerous anticipated M2M / IoT applications across all sectors, the
throughput requirement for each anticipated application also vary. Based on the trade-off between
the energy gains from mainly communication energy cost and computing energy cost when CS
algorithm is applied to the M2M communication devices, it is imperative to ascertain the
throughput at which the applications of CS algorithm are energy efficient.
The 𝐸𝑡𝑥>𝐸𝑟𝑥 for a typical wireless ad hoc network and based on the fact that the wireless device
to be used for M2M communications / IoT are expected to be dumped in order to elongate the life
span of the devices [29]. It can be deduced that most of the M2M / IoT communication devices
will rarely receive packets except for updates in terms of the available spectra for
communication. Hence, in this EECST model, the assumptions made are summarized as follows;
a) M2M communication cellular structure as proposed by Weightless Special Interest Group
[30], hence, the major communication cost will be on transmitting the data packets to the
base station, which do most processing and scheduling in the M2M communication
networks.
b) The battery power of the devices is very limited, hence the need to know the throughput at
which the applications of CS algorithm is energy efficient.
c) The channels of communication are sub 1GHz spectra (902 – 928) MHz ISM Band, hence
have limited bandwidths as proposed in [31].
d) The data processing is done on the base station
e) Multipath fading does not exist
f) Binary phase shift keying (BPSK) signal is used in the model
Let the amount of energy required to transmit a single bit of data be 𝒆𝒕𝒃, which is equivalent to
the number of clock cycles required to transmit a single bit of data 𝒏 and let the energy cost per
clock cycle be 𝒆𝒄𝒄. To transmit x bit(s) of data without compression, the 𝑬𝒕𝒙 is evaluated as given
in (45). As mentioned above, the application of CS algorithm increases the energy cost of
computing / processing. Let the number of clock cycle required to execute CS algorithm on the
data sample be 𝐶𝑛, and the 𝑬𝒕𝒙 when CS algorithm is applied is given in equation (45).
𝑬𝒕𝒙 = 𝒙 × 𝒆𝒕𝒃 ≡ 𝒙 × 𝒏 × 𝒆𝒄𝒄 (45)
For the benefit of the context of this discussion, it is imperative to split the total number of bits to
be transmitted into the preamble which is assumed to be constant for both compressed and
uncompressed signal. Let the number of bits in the preamble be 𝒙𝟏 and the remaining number of
bits in the packet be 𝒙𝟐, putting the above assumptions in (45) will result to equation (46).
𝑬𝒕𝒙𝟏 = (𝒙𝟏 + 𝒙𝟐) × 𝒆𝒕𝒃 ≡ (𝒙𝟏 + 𝒙𝟐) × 𝒏 × 𝒆𝒄𝒄 (46)
To be able to evaluate 𝑬𝒕𝒙 when CS algorithm is implemented on the M2M communications
device, let the number of clock cycle required to perform data compression using CS algorithm
be 𝒏𝒄 and the percentage of compression be ɳ, then the 𝑬𝒕𝒙 using CS algorithm is given in
equation (47).

87
𝑬𝒕𝒙𝟐 = 𝒙𝟏 × 𝒆𝒕𝒃 + ɳ × 𝒙𝟐 × 𝒆𝒕𝒃 + 𝒏𝒄 × 𝒆𝒄𝒄 ≡ (𝒙𝟏 + ɳ × 𝒙𝟐) + 𝒏𝒄 × 𝒆𝒄𝒄 (47)
Equation (46) and (47) above are suitable for single – input to single – output (SISO) form of
communication. For multiple input multiple output (MIMO) form of communication, the above
equations will not be suitable. This is as a result of the increase in circuit complexity which will
increase the computing / processing energy cost, coupled with ℵ number of bits that is being
transmitted simultaneously. However, Zimran R et al [32] stated based on their analysis that,
using MIMO for data transmission is more energy efficient for transmission across long distance.
While across short distance, 𝑬𝒄𝒐𝒎𝒑𝒖𝒕𝒂𝒕𝒊𝒐𝒏 is more than 𝑬𝒕𝒙 as a result of the circuit complexity.
In order to evaluate 𝑬𝒕𝒙 when MIMO is used, let the energy cost of transmitting ℵ bits of data
using MIMO per transmission be 𝒆ℵ , and the number of clock cycles required by a MIMO
circuit to transmit x bits of data be 𝒏ℵ,then the energy cost of transmitting x bits of data is as
given in equation (48).
𝑬𝒕𝒙𝟑 = (
𝒙𝟏 + 𝒙𝟐
ℵ
) × 𝒆ℵ ≡ (
𝒙𝟏 + 𝒙𝟐
ℵ
) × 𝒏ℵ × 𝒆𝒄𝒄 (48)
Then the 𝑬𝒕𝒙 on the application of CS algorithm is as given in equation (49).
𝑬𝒕𝒙𝟒 = (
(𝒙𝟏 + ɳ × 𝒙𝟐)
ℵ
) × 𝒆ℵ + 𝒏ℵ × 𝒆𝒄𝒄 ( 49)
Considering SISO scenario as given in (46) and (47), let’s take the energy cost of MSP430 serial
micro-processor in active state as the yardstick for evaluation, in which 𝑒𝑡𝑏 =
230𝑛𝑗 and 𝑒𝑐𝑐 = 0.729𝑛𝑗 for a WSN with a range of 100 meters [22]. Also as mentioned
earlier, the energy cost of processing CS algorithm in the node(s) is determined by the number of
clock cycles required by the devices to perform the compression operations. Table 1 shows the
number of clock cycles required by MSP430 micro-processor to perform certain floating-point
operations.
Table 1. The number of clock cycles for specific floating-point operations [22]
FLOATING POINT OPERATION(S) NUMBER OF CLOCK – CYCLES
Addition 184
Subtraction 177
Multiplication 395
Division 405
Comparison 37
The type of the CS algorithm and the size of the sensing matrix will determine the number of
clock cycle required by the micro-processor to perform the compression operation on the device.
For instance, a micro-processor with an impeded Sub-threshold (Sub-VT) CS processor will
require 8460 clock cycles to apply 50% compression on 512 samples of ECG data [33]. The A in
(18) is constructed by 12 random indices per column, and the sampling rate of the signal is
125Hz. Also, assuming that the length of the MAC addresses for both the compress and un-
compressed signal samples are same, then only 𝑥2 in (46) and (47) is considered in the
computation. Figure 5 shows the data rates at which compression becomes energy efficient.

88
Based on the input variables as stated, the throughput at which compression becomes energy
efficient is at 53.63 ≈ 54 bits. Though the number of clock cycles used for compression is based
on Sub-threshold (Sub-VT) CS processor, the number of clock cycles required for performing CS
compression algorithm varies from one micro-controller to another. This is because of the
variations in the inherent properties of various micro-controllers like speed, energy requirement
per clock cycle, etc. and the variations on the various computation requirements of various CS
algorithms.
Consequent to the variations in the capabilities of different microprocessor, and the number of
clock cycles required to perform CS algorithms, Figure 5 shows the minimum data rates that are
required for the application of CS algorithm on M2M / IoT communication devices to be energy
efficient. Figures 5 and 6 are derived from equation (46) and (47) which can be used for
evaluating the data rates in a BPSK scenario, while equation (48) and (49) can be used in
evaluating the data rates at which the applications of CS algorithm on M2M / IoT communication
devices become energy efficient in MIMO scenarios. As can be seen in figure 6, when the
number of clock cycles is increased to 20000, the data rate at which compression becomes energy
efficient is 127 bits and at 50000, the data rate becomes approximately 317 respectively.
Figure 5. The CS compressed Vs uncompressed transmitted energy cost.

89
Figure 6. The CS compressed Vs uncompressed transmitted energy cost @ n = 20,000 and 50,000
7. CONCLUSION
The EECST model has become handy for M2M / IoT communication experts to use in evaluation
based on the throughput requirement of various M2M / IoT communication applications, the rates
at which the application of the CS algorithm on the wireless devices becomes energy efficient.
The higher the computational energy cost for executing CS algorithm on the wireless device, the
higher the throughput at which the application of CS algorithm becomes energy efficient. The
values used for etbis based on WSN with a range of 100 meters, the value is more in the range of 4
kilometres. Though, it is imperative to note that the application of CS on any of the applications is
subject to the recoverability error performance of such signal on the receiver. Also, the type of the
sensing matrix used for compression, and the type of compressed signal recovery algorithm is
very vital in determining the difference between the original signals, and recovered signals.
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