Design of a cosine pulse transceiver operating in the discrete-time domain

: Introduction/purpose: The paper presents the theory and the design issues of a transceiver for a discrete-time cosine pulse transmission. The operation of the transceiver and all signals are analyzed in both time and frequency domains. Methods: The detailed theoretical models of the transmitter and the receiver are presented based on their block schematics expressed in terms of mathematical operators. The transceiver blocks are simulated and the results of their simulation are presented and compared with the theoretical results. Results: Discrete-time signals at the input and the output of each transceiver block are derived in the mathematical form and presented in the time and frequency domain. A transceiver simulator is developed in Matlab. The simulation results confirmed the theoretical findings. Conclusion: The results of this work contribute to the theoretical modeling and design of modern transceivers that can be used for discrete-time cosine pulse transmission.


Introduction
Designs of communication transmitters and receivers are based on the presentation of signals in the continuous-time domain, i.e., in their analysis and synthesis, signals are presented as continuous functions of time. Related communication systems, composed of these devices, are known under the name of digital communication systems (Haykin, 2014;Proakis, 2001). Due to the development of advanced digital technology, signals of modern communication systems are represented by discrete- The coherent receiver will demodulate the received band-pass (BP) signal using a low-pass filter (LPF) and generate an estimate of the cosine pulse or the rectangular pulse transmitted. Firstly, the modulated signal s(n) is multiplied by the carrier sc(n) to obtain the demodulated cosine pulse scm(n). Then, the cosine pulse is multiplied by the LF cosine signal sm(n) to obtain a signal smm(n) that contains the rectangular pulse that can be extracted by the low-pass filter (LPF). In the case of the system simulation, a band-pass noise generator should be used to investigate the operation of the system in real conditions. Because the system operates with discrete-time signals, the noise generator needs to generate a BP discrete-time noise that will be added to the modulated discrete-time signal. 562 VOJNOTEHNIČKI GLASNIK / MILITARY TECHNICAL COURIER, 2023, Vol. 71, Issue 3 Transmitter operation We will generate the discrete-time cosine pulse and then modulate the carrier with that cosine pulse. The block schematic of the transmitter is presented in Figure 1. The generated pulse can be considered an LF cosine pulse. However, our question is how to generate the band-pass sinusoidal pulse with the modulating signal that is this LF cosine signal. For that purpose, as we expect, the carrier frequency should be much higher than the bandwidth of the LF cosine pulse. Let us assume, for the sake of explanation and presentation, that the carrier frequency is 10 times higher than the middle frequency of the LF cosine pulse.
To generate this modulated pulse, we need to multiply the LF cosine pulse with the carrier, as shown in Figure 1. The carrier frequency is 10 times higher than the frequency of the cosine pulse, i.e., fc = 10fm. Suppose the minimum number of samples in one period of the carrier is Nc = 4. Therefore, the number of samples N of the rectangular pulse p(n) (for 2 oscillations of the LF cosine pulse and 20 oscillations of the carrier and the number of samples Nm in one period of the LF cosine signal) needs to be calculated to accommodate the 10 times higher frequency of the carrier, i.e., 10 2 20 4 80 (1) The number of samples inside one period of the LF cosine pulse now is calculated to be / 2 40 m NN  . Therefore, to perform the multiplication of the rectangular pulse by the LF cosine signal and then modulate the carrier, the number of their samples should be 80. The mathematical expressions and related wave shapes of the transmitter signals will be presented in the following analysis. For the calculated values of the discrete signals, we can analyze their properties in the time and frequency domain at each block of the transceiver. The rectangular pulse presentation in the time and frequency domain. Based on the calculated number of samples inside the processed signals, the graphs of the discrete-time (dt) rectangular pulse p(n) in the discretetime domain are presented in Figure 2. The signal is expressed in terms of the Kronecker delta functions as a convolution of the signal and the delta functions. i.e.,  It is to note that the discrete-time pulse values are defined for each whole number n and have no values in the intervals between neighboring numbers. We can say that the signal values do not exist in these intervals, even though these intervals are used to process the discrete signal values. The related magnitude and phase spectral densities can be obtained from the DTFT of the dt rectangular pulse which is defined as and is to be calculated for N = 80. The magnitude spectral density can be expressed as   | | 2 , 0, 1, 2,3,...
which is graphically presented in Figure 3 for the defined duration of the pulse N = 80 and the amplitude A = 2. The zeros in the magnitude spectrum occur for the condition expressed as On the other hand, the energy spectral density (ESD) of the pulse is defined as the magnitude spectral density square and expressed as In the frequency domain, the energy of the pulse is calculated as an integral of the ESD in this way (Integral calculator, 2023)   2 2 2 2 2 2 1 sin( / 2) 160 The ESD can be calculated also as the DTFT of the autocorrelation function of the discrete rectangular pulse (Berber, 2019(Berber, , 2021.
The LF cosine signal in the time and frequency domain. The LF cosine signal has Nm = 40 samples per oscillation, which can be understood as a subcarrier in the system. Therefore, its frequency is which is shown in a graphical form in Figure 4a).

Figure 4 − a) Waveshape of the LF sinusoid signal, b) related amplitude spectral density Рис. 4 -а) Форма волны низкочастотного синусоидального сигнала, б) соответствующая амплитудная спектральная плотность Слика 4 -а) Таласни облик НФ синусоидалног сигнала, б) односна спектрална густина амплитуде
The signal in the frequency domain can be directly found for any Nm simply applying Euler's formula on the time domain as follows or the amplitude spectral density expressed as which is a periodic function of the continuous frequency Ω with the period of 2π, as presented in Figure 4b). This spectrum can be represented by a periodic stream of the Dirac delta functions (Papoulis & Pillai, 2002) weighting π at periodic frequencies and zeros everywhere else.
The power of the signal in the time domain is calculated as This is a power signal (Cavicchi, 2007;Berber, 2021). Therefore, its average power is to be calculated in the infinite interval, according to this expression Because this signal in the frequency domain is a periodic function of the continuous frequency Ω with a period of 2π, it can be represented by a periodic stream of the Dirac delta functions, as presented in Figure 4b).
The energy of the signal is The infinite energy value can be confirmed by its calculation in the time domain as 2 1 2 lim cos ( ) lim (1 cos / 2) lim 2 2 a a sm m a a a n a n a a E n n The LF cosine pulse in the time and frequency domain. The cosine pulse m(n) is obtained by multiplying the sm(n) shown in Figure 1 by the rectangular pulse p(n). This is again a cosine function. However, it is not a periodic function as it was the LF cosine signal sm(n). Consequently, it will not be expressed in the frequency domain in terms of the Dirac delta functions. Instead of one spectral component, the spectrum of the cosine pulse will have a bandwidth around the frequency of the periodic cosine signal. The cosine pulse can be expressed in the time domain as Design of a cosine pulse transceiver operating in the discrete-time domain, pp.559-587 and, based on the modulation theorem, in the frequency domain as where its shifted components are expressed as The cosine pulse and the related magnitude spectral density are presented in Figure 5.
The amplitude spectral density is a periodic function of continuous frequency with the period of 2π as presented in Figure 5b). The signal is an energy signal, and its energy can be calculated from the energy spectral density (Integral calculator, 2023) as The carrier in the time and frequency domain. For the defined frequency, the carrier of a unit amplitude is expressed in the discrete-time domain as as presented in Figure 6a). The carrier in the frequency domain can be directly found, for the number of samples Nc = 4 in one period of the carrier, by simply applying Euler's formula on the time domain expression of the signal and we may have which is a periodic function of frequency with the period of 2π, as presented in Figure 6b). Because this signal is a periodic function of the continuous frequency Ω with a period of 2π, it is represented by a periodic stream of the Dirac delta functions (Papoulis & Pillai, 2002), as presented in Figure 6b). Design of a cosine pulse transceiver operating in the discrete-time and presented in Figure 7a). By applying the modulation property of DTFT, we may get the amplitude spectral density of that signal as We can find its shifted components in (24) where N = 80 for our case. Then, based on expression (17) for the amplitude spectral density of m(n), the magnitude spectral density of the modulated signal (24) finally is , the magnitude spectral density of the modulated signal is presented in Figure 7b). The magnitude spectrum is a periodic function with a period of 2. The two-sided spectrum of the signal can be investigated inside the bandwidth around the carrier frequency of /2.
which can be calculated as shown before. From the energy expression, it is easy to find the power of the signal and vice versa.

Simulation of the transmitter operation
We performed a simulation in Matlab of the transmitter presented in Figure 1 (Ingle & Proakis, 2012).
The signals are generated in the time and frequency domain and presented in graphical forms. A simulated LF cosine pulse in the time and frequency domain is shown in Figure 10. The cosine pulse m(n) has the amplitudes A = 2 in the interval from n = 0 to n =80. It is not a periodic function, and its spectrum is a shifted version of the spectrum of the rectangular pulse to the frequency Ωm. The graphs in Figure 10 correspond to the theoretical graphs shown in Figure 5. The simulated high-frequency carrier is shown in the time and frequency domain in Figure 11. The processed carrier was in the interval from n = 0 to n = 800. In Figure 11, only a part of the signal is shown for the sake of understanding its shape. These graphs correspond to the theoretically expected graphs presented in Figure 6.
The simulated modulated signal in the time and frequency domain is presented in Figure 12. The same signal in the time domain is presented in an extended form in Figure 13. These graphs correspond to the theoretically expected graphs presented in To see the shape of the signal, the modulated signal s(n) is presented in the whole interval from n = 0 to n = N-1 in Figure 13.

Pulse demodulator
The demodulation of the discrete modulated signal s(n) results in the discovery of the modulating cosine and the rectangular pulse. The receiver blocks involved in the procedure of received signal demodulation are presented in Figure 1.
The output signal of the demodulator multiplier. Firstly, the received discrete modulated signal is multiplied by the carrier to get the signal The signal can be expressed in this form in the time domain Therefore, the DTFT of this signal gives us its amplitude spectral density expressed as where the first component M(Ω) is the LF component containing the spectrum of the modulating cosine pulse and its shifted spectrum to the doubled carrier frequency. The LF component can be expressed as where the shifted components are To obtain the multiplied signal () cm S  expressed by (30), we can calculate the shifted spectral components as follows and is presented in Figure 14. The power and energy of the signal in the time domain can be calculated from (29) and expressed as Using the modulation theorem, the amplitude spectral density of this signal will be expressed as a shifted version of the spectrum in (34)  The calculated spectrum contains an LF part around the zero frequency, which contains the spectrum of the modulating rectangular pulse. We can use an LP filter to extract this signal. This operation will be approximate, meaning that a part of the spectrum of the neighboring components to the pulse spectrum will be added to the signal causing distortion. Also, the pulse spectrum will be reduced to its two arcades which will reduce the power of the signal. We will take this reduction of the power in the following considerations and calculations.
LPF operation in the time and frequency domain. An ideal LP filter with the gain Hd is used to eliminate the HF components in this spectrum and obtain the demodulated signal representing the rectangular pulse p(n) that was sent by the transmitter. The filter eliminated all high-frequency components and the result is the LF pulse Design of a cosine pulse transceiver operating in the discrete-time domain, pp.559-587 where the filter transfer characteristic is rectangular as shown in Figure 15 by a dotted graph of the amplitude Hd. If there is no attenuation of the filter, i.e., Hd =1, an approximation of the output spectrum can be obtained as shown in Figure 16. Due to the limitation of the spectrum of the demodulated pulse p(n) at the receiver side, the received pulse at the output of the LP filter will not be rectangular as the simulation will confirm. In the time domain, the output signal of the filter, md(n), will be a convolution of the filter input signal smm(n) and the impulse response of the filter hd(n), as will be shown in the simulation. The result of this convolution is the pulse md(n), which is a distorted version of the rectangular pulse.
Calculation of the system attenuation. The total value of the demodulated rectangular pulse energy can be calculated from the pulse spectrum in Figure 15, assuming an all-pass LF filter and ideal filtering, as This energy of the signal is much smaller than the energy of the pulse at the transmitter side, which was NA 2 , as calculated in (6) The power attenuation of the received signal is To receive the pulse with the required power, we need to use amplifiers in the receiver. These amplifiers will compensate for the loss of power caused by the explained signal processing. However, due to the propagation of the signal, there will exist additional attenuation that needs to be compensated by the amplification inside the receiver. Finally, as shown in Figure 1, there will be a noise present in the channel that is added to the signal. The noise influence on the signal transmission needs to be also considered, which is a separate problem in the analysis of the system.
If we are interested in the detection of the phase of the transmitted cosine pulse, we can use a correlator on the receiver side. In that case, the polarity of the correlator output will give us evidence about the phase of the transmitted pulse. Simulation of the receiver The receiver operation is simulated in Matlab. The simulated operation of the first modulation multiplier is presented by the graphs of its output discrete-time signal scm(n) in the time and frequency domain in Figure 17, which was theoretically analyzed and presented in the frequency domain in Figure 14. This signal is multiplied by an LF signal sm(n) to get the signal smm(n) that contains a low-frequency component that represents the modulating signal. The simulated signal in the time and frequency domain is shown in Figure 18. The corresponding signal, theoretically calculated, is presented in Figure 15. The LF signal smm(n) is processed in the LP filter to get the LF modulating signal. For the filter transfer characteristic shown in Figure 14, the impulse response hd(n) is calculated.
Then the convolution of this impulse response and the input signal smm(n) is performed in the time domain as shown in Figure 19.
The LPF is a linear time-invariant discrete-time system, and this convolution can be performed.
The result of the convolution is the positive pulse md(n) that is generated at the output of the receiver and corresponds to the rectangular pulse sent at the transmitter side.
This pulse obtained by simulation is presented in Figure 20. Conclusions This paper presented the theoretical model and the simulation results of a communication system analysis for a cosine pulse transmission. A detailed block schematic of the system's transmitter and receiver is presented in the form of mathematical operators and all input-output signals are presented in both time and frequency domains using exact mathematical expressions. To calculate the attenuation of the signals in the system, the powers and the energies of the signals are calculated for all signals processed in the system.

584
It is shown that the application of a low-pass filter inside the receiver allows the detection of the modulating signal, despite its distortion due to the processing in the system. The signals in the analyzed blocks of the transceiver are processed in the discrete-time domain. All theoretical results are confirmed by simulations.