Adaptive time-frequency multiplexing for 5G applications

Journal Pre-proofs Regular paper Adaptive Time-Frequency Multiplexing for 5G Applications Mohsen Farhang, Hossein Khaleghi Bizaki PII: DOI: Reference:

S1434-8411(19)32412-4 https://doi.org/10.1016/j.aeue.2020.153089 AEUE 153089

To appear in:

International Journal of Electronics and Communications

Received Date: Accepted Date:

24 September 2019 18 January 2020

Please cite this article as: M. Farhang, H. Khaleghi Bizaki, Adaptive Time-Frequency Multiplexing for 5G Applications, International Journal of Electronics and Communications (2020), doi: https://doi.org/10.1016/ j.aeue.2020.153089

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Adaptive Time-Frequency Multiplexing for 5G Applications Mohsen Farhang1, Hossein Khaleghi Bizaki2* Malek-Ashtar University of Technology [email protected], [email protected]

*

Corresponding Author

1

Abstract- In this paper, we provide an optimization framework for channel-adaptive pulse and lattice scaling in a general Time-Frequency (T-F) multiplexing scheme. Closed form relations for the optimum pulse and lattice scaling are derived analytically considering the requirements of two important 5G applications, i.e. tactile internet and cognitive radio. For tactile internet, we solve the problem with a constraint on the time dispersion of the pulse shaping filter at the receiver in order to decrease the length of Cyclic Prefix (CP) and consequently improve the spectral efficiency. Then, aiming for limiting the Out-Of-Band (OOB) leakage in cognitive radio applications, we find a solution to the optimization problem with a constraint on frequency dispersion of pulse shaping filter at the transmitter. Finally, we consider both of these constraints and solve the problem analytically. The results show that using optimum pulse and lattice scaling, improves the Signal to Interference Ratio (SIR) significantly, compared to nonoptimal pulses and lattices. Furthermore, it is shown that applying the constraint to either Receive (Rx-) filter time dispersion or Transmitter (Tx-) filter frequency dispersion does not degrade the SIR significantly while limiting both of them leads to a mismatch between pulse shaping filters and degrades the SIR considerably. Keywords: time-frequency multiplexing, adaptive communications, 5g network, pulse and lattice scaling, cognitive radio, tactile internet.

1. Introduction Development from 1G to 4G has been driven by the need for higher data rate and lower latency. In response to foreseen explosion of data, the next generation, i.e. 5G, is also progressing the same path but with added requirements such as flexibility [1-3]. Various applications and services in 5G force diverse requirements on the system design, ranging from relaxed synchronization for Internet of Things (IoT), to ultra-low latency for Machine Critical Communications (MCC). Hence, 5G is expected to provide much more adaptive, flexible, and efficient radio access technologies compared to today's systems [4, 5]. In this regard, just as 4G-LTE already benefits from channel-dependent scheduling, link adaptation, and hybrid Automatic Repeat reQuest (ARQ) [6], more aspects of adaptation seem to be utilized in the 5G physical layer [7]. A comprehensive overview of the most important 5G physical layer aspects is available in [8]. Channel-adaptive communication systems exploit the Channel State Information (CSI) in order to optimally adapt their transmit/receive specifications regarding the channel characteristics [9, 10]. In multicarrier communication systems, link adaptation is a key technique to realize a high data rate and reliable transmission over time-frequency dispersive channels. Link adaptation provides many parameters that can be adjusted relative to the CSI, including modulation type, data rate, transmit power, coding rate or scheme, pulse shape and T-F lattice [10, 11], where the latter two cases are subjects of this paper. A pulse shape determines the density of the symbol energy in time and frequency domain and has an important effect on the dispersion attributes of the signal. Pulse adaptation refers to adjustment of the pulse shape regarding the channel dispersion in time and frequency. The lattice is a set generated by sampling the T-F plane and defines the coordinates of pulse shaping filters in T-F grid. Lattice adaptation is equivalent to adapting the symbol spacing in time and subcarrier spacing in frequency, to the channel delay and Doppler spread [12, 13]. Generally, matching the proportion of the channel dispersion to the pulse dispersion in the time and frequency leads to maximum robustness of the multicarrier schemes against doubly-dispersive channels [12, 14]. In [15] the pulse adaptation problem is studied without any theoretical explanations. Several iterative algorithms are presented in [16] for optimization of pulse and lattice with respect to the scattering function of the Wide Sense Stationary Uncorrelated Scattering (WSSUS) channel. In [14, 17] the lattice is constructed with a hexagonal shape, instead of rectangular, in order to achieve better protection against InterSymbol Interference (ISI) and InterCarrier Interference (ICI). In [18] the pulse shape adaptation in OFDM/OQAM system is studied with focus on the Extended Gaussian Functions (EGF). The problem of designing transmit and receive pulse shapes in order to decrease ISI and ICI, is considered in [19]. In [20] non-orthogonal pulses are considered and adapted to realistically available a priori knowledge of the doubly dispersive channel characteristics. In [21, 2

22] the pulse and the lattice adaptation is applied for multiple users. In [16] assuming WSSUS channel, the pulse and lattice scaling rules for SIR maximization are derived analytically by a twostep procedure including (1) gain maximization and (2) interference minimization. To the best knowledge of authors, this problem is not studied considering the constraints forced by different application requirements yet. The main contribution of this paper is the derivation of optimal pulse and lattice scaling rules by solving SIR maximization problem under different constraints dictated by requirements of two 5G applications, i.e. cognitive radio [23] and tactile internet [24]. In this regard, analytical expressions are derived for the cross-ambiguity function of Gaussian pulses and also interference power in a T-F resource element arising from other subcarriers and subsymbols. The optimization problem is formulated and solved analytically, with a constraint on the Rx-filter time dispersion in order to have an adequately short CP and, as a result, improve the spectral efficiency which is crucial in tactile internet. The same approach is used to limit the Tx-filter frequency dispersion in order to reduce the OOB leakage which is an important issue in 5G cognitive radios. Finally, the SIR maximization is solved considering limits on both Tx- and Rx-filter dispersion in frequency and time, respectively. The remainder of this paper is organized as follows: Section 2 provides the system model. The general optimization problem is formulated and solved in section 3. The constraints for different 5G scenarios are applied in Section 4. Results are given in Section 5 and conclusions are drawn in Section 6.

2. System Model Any multicarrier transmission in its general form can be considered as a T-F multiplexing scheme. The baseband transmitted signal is given by [16]: 𝑠(𝑡) =

∑

𝑥𝑚,𝑘𝛾𝑚,𝑘(𝑡) =

(𝑚,𝑘)𝜖ℤ2

∑

𝑥𝑚,𝑘S𝑚𝑇,𝑘𝐹[𝛾(𝑡)],

(1)

(𝑚,𝑘)𝜖ℤ2

in which 𝑥𝑚,𝑘 is the symbol being transmitted on the 𝑚𝑡ℎ subsymbol and 𝑘𝑡ℎ subcarrier, with the property 𝐸{𝒙𝒙 ∗ } = 𝐼, where 𝒙 = (…,𝑥𝑚,𝑘,…)𝑇. S is the shift operator and 𝛾𝑚,𝑘(𝑡) is a T-F shifted version of transmitter pulse shaping filter 𝛾(𝑡) so that: S𝑚𝑇,𝑘𝐹[𝛾(𝑡)] ≝ 𝛾𝑚,𝑘(𝑡) = 𝛾(𝑡 ― 𝑚𝑇)𝑒𝑖2𝜋𝑘𝐹𝑡,

(2)

where T and F are symbol period and subcarrier spacing, respectively. We apply a random linear operator H with kernel 𝐻(𝜏,𝜈) in order to model the baseband T-F doublydispersive channel as [14]: H[𝑠(𝑡)] =

∫

𝐻(𝜏,𝜈)𝑠(𝑡 ― 𝜏)𝑒𝑖2𝜋𝜈𝑡𝑑𝜏𝑑𝜈.

2

(3)

ℝ

The kernel 𝐻(𝜏,𝜈) is the delay-Doppler spread function and equals to the Fourier transform of the time-varying impulse response of the channel ℎ(𝑡,𝜏) with respect to 𝑡. Generally, the statistical characteristics of the channel satisfy the Wide Sense Stationary Uncorrelated Scattering (WSSUS) assumption, which implies that uncorrelated delays and Doppler shifts. Hence, we have [16]: 𝐸{𝐻(𝜏,𝜈)𝐻 ∗ (𝜏′,𝜈′)} = 𝑆𝐻(𝜏,𝜈)𝛿(𝜏 ― 𝜏′,𝜈 ― 𝜈′),

(4)

where 𝑆𝐻(𝜏,𝜈) is the channel scattering function that determines the second order statistics of the WSSUS channel [14]. In this paper, without loss of generality, 𝐻(𝜏,𝜈) is assumed to have zero mean and unit variance, which indicates no overall path loss. Considering the Additive White Gaussian Noise (AWGN) process 𝑛(𝑡), the received signal can be denoted as, 𝑟(𝑡) = H[𝑠(𝑡)] + 𝑛(𝑡) (5)

3

The data symbol 𝑥𝑚,𝑘 is obtained by the projecting 𝑟(𝑡) on T-F shifted version of receiver pulse shape 𝑔(𝑡), i.e., 𝑔𝑚,𝑘(𝑡) = 𝑔(𝑡 ― 𝑚𝑇)𝑒𝑖2𝜋𝑘𝐹𝑡, hence we have 𝑥𝑚,𝑘 =< 𝑔𝑚,𝑘,𝑟 >=

∫

∗ 𝑔𝑚,𝑘 (𝑡)𝑟(𝑡)𝑑𝑡

(6)

2

ℝ

The projected noise power is independent of (𝑚,𝑘) and is given by 𝜎2≔𝐸𝑛{| < 𝑔,𝑛 > |2} The signal and interference power are given by:

(7)

𝑃𝑠𝑖𝑔 = |𝐻(𝑚,𝑘),(𝑚,𝑘)|2

∑

𝑃𝑖𝑛𝑡 =

(8) 2

|𝐻(𝑚,𝑘),(𝑖,𝑗)| ,

(9)

(𝑚,𝑘) ≠ (𝑖,𝑗)

in which 𝐻(𝑚,𝑘),(𝑖,𝑗) =< 𝑔𝑚,𝑘,H[𝛾𝑖,𝑗] >=

∫

𝐻(𝜏,𝜈) < 𝑔𝑚,𝑘,S𝜏,𝜈[𝛾𝑖,𝑗] > 𝑑𝜏𝑑𝜈.

(10)

> |2𝑆𝐻(𝜏,𝜈)𝑑𝜏𝑑𝜈

(11)

ℝ2

Thus, the average signal and interference power are:

∫ | < 𝑔,S ∑ ∫ | < 𝑔,S

𝐸𝐻{𝑃𝑠𝑖𝑔} = 𝐸𝐻{𝑃𝑖𝑛𝑡} =

2

ℝ

> |2𝑆𝐻(𝜏,𝜈)𝑑𝜏𝑑𝜈.

𝜏 + 𝑚𝑇,𝜈 + 𝑘𝐹 [𝛾]

ℝ2

(𝑚,𝑘) ≠ (0,0)

𝜏,𝜈[𝛾]

(12)

Using the cross-ambiguity function of 𝑔 and 𝛾, given by 𝐴𝑔𝛾(𝜏,𝜈) =< 𝑔,𝑆𝜏,𝜈 𝛾 > , we have 𝐸𝐻{𝑃𝑠𝑖𝑔} = 𝐸𝐻{𝑃𝑖𝑛𝑡} =

∑ ∫

(𝑚,𝑘) ≠ (0,0)

∫

|𝐴𝑔𝛾(𝜏,𝜈)|2𝑆𝐻(𝜏,𝜈)𝑑𝜏𝑑𝜈 = 𝐼(0,0),

|𝐴𝑔𝛾(𝜏 + 𝑚𝑇,𝜈 + 𝑘𝐹)|2𝑆𝐻(𝜏,𝜈)(𝜏,𝜈)𝑑𝜏𝑑𝜈 =

2

ℝ

(13)

ℝ2

∑

𝐼(𝑚𝑇,𝑘𝐹) (14)

(𝑚,𝑘) ≠ (0,0)

in which 𝐼(∆𝑇,∆𝐹) is the interference power between symbols with distance ∆𝑇 and ∆𝐹 in time and frequency, respectively given by: 𝐼(∆𝑇,∆𝐹) =

∫

|𝐴𝑔𝛾(𝜏 + ∆𝑇,𝜈 + ∆𝐹)|2𝑆𝐻(𝜏,𝜈)𝑑𝜏𝑑𝜈

2

ℝ

(15)

The optimal time-frequency signaling in terms of SIR is now given as the solution to the problem: 𝐸𝐻{𝑃𝑠𝑖𝑔} 𝐼(0,0) 𝑎𝑟𝑔𝑚𝑎𝑥 = 𝑎𝑟𝑔𝑚𝑎𝑥 , (16) 𝑘 𝑔,𝛾,𝑇,𝐹 𝐸𝐻{𝑃𝑖𝑛𝑡} 𝑔,𝛾,𝑇,𝐹 ∑ 𝐼(𝑚𝑇, ) (𝑚,𝑘) ≠ (0,0) 𝜀𝑇 with below constraints applied to pulses and lattice structure as: C1) Normalized pulses, i.e. ∥ 𝑔 ∥ 2 = ∥ 𝛾 ∥ 2 = 1. C2) Constant spectral efficiency, i.e. 𝑇𝐹 = 𝜖 ―1 = 𝑐𝑜𝑛𝑠𝑡. In the next section, the above problem is solved assuming both channel scatter function and pulse shaping filters being Gaussian.

3. General Optimization Problem Consider pulse shaping filters at transmitter and receiver to be Gaussian as: 1

( ) 2𝑏 𝑔(𝑡) = ( ) 𝑒 𝜋

2𝑎 4 ―𝑎𝑡2 𝛾(𝑡) = 𝑒 𝜋 1 4

4

―𝑏𝑡2

,

(17)

(18)

where 𝑎 and 𝑏 are parameters used for adjusting the pulse dispersion in the time and frequency at the transmitter and receiver filters, respectively. The cross-ambiguity function is derived for 𝑔(𝑡) and 𝛾(𝑡) (see Appendix A): 2

2 2

𝑎𝑏𝜏 + 𝜋 𝜈 2 𝑎𝑏 ―2( 𝑎 + 𝑏 ) 2 |𝐴𝑔𝛾(𝜏,𝜈)| = 𝑒 𝑎+𝑏

(19)

𝜋

𝛼 ― 𝛼(𝜏2 + 𝜈2) 2 2𝑒

The symmetric Gaussian channel scattering function given in [25] can be written to a general asymmetric form as: 𝜋 𝛼𝛽 ― 2(𝛼𝜏2 + 𝛽𝜈2) 𝑆𝐻(𝜏,𝜈) = , 𝜏 > 0, (20) 𝑒 2 where 𝛼 and 𝛽 are real and positive parameters that determine the delay and Doppler spread, respectively. Assuming Gaussian channel scattering and pulse shapes, we have (see Appendix B): ― 𝑥𝑦𝛼𝛽 𝜋 𝜋 𝑒 𝐼(∆𝑇,∆𝐹) = 4 𝑥+ 𝛼 𝑦+ 𝛽 2 2

(

2𝑎𝑏

)(

)

(

𝑥―

) (

)

𝑥2 𝑦2 ∆𝑇2 ― 𝑦 ― ∆𝐹2 𝜋 𝜋 𝑥+ 𝛼 𝑦+ 𝛽 2 2

, (21)

2𝜋2

in which 𝑥 = 𝑎 + 𝑏 , 𝑦 = 𝑎 + 𝑏. 1

Now with replacing 𝐹 with 𝜀𝑇 from condition C2, the optimization problem of (15) can be written as 𝐼(0,0) (𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) = 𝑎𝑟𝑔𝑚𝑎𝑥 . (22) 𝑘 𝑇,𝑎,𝑏 ∑ 𝐼(𝑚𝑇, ) (𝑚,𝑘) ≠ (0,0) 𝜀𝑇 Usually we have 𝜏𝑟𝑚𝑠𝑓𝑑 ≪ 1 for practical wireless WSSUS channels, and the conditions 𝜏𝑟𝑚𝑠 < 𝑇 and 𝑓𝑑 < 𝐹 are satisfied. Hence, without considerable performance loss, considering only the interference from the neighboring symbols is sufficient [14]. Therefore, in what follows, we only consider the following pairs in interference term: (𝑚,𝑘) ∈ {(1,0),( ―1,0),(0,1),(0, ― 1),(1,1),(1, ― 1),( ―1,1),( ―1, ― 1)}. Figure 1 depicts the interference from neighbor symbols. Noting that 𝐼(𝑚𝑇,𝑘𝐹) is an even function regarding 𝑚 and 𝑘.

𝐼(0,𝐹) 𝐼(𝑇,0) 𝐼(𝑇,𝐹)

Figure 1. Interference from neighbor symbols in T-F lattice

Hence, the problem simplifies to:

5

(𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) = 𝑎𝑟𝑔 𝑚𝑎𝑥 𝑇,𝑎,𝑏

= 𝑎𝑟𝑔 𝑚𝑎𝑥 𝑇,𝑎,𝑏

𝐼(0,0) {2𝐼(𝑇,0) + 2𝐼(0,𝐹) + 4𝐼(𝑇,𝐹)} 1

𝑥2 ―(𝑥 ― )𝑇2 𝜋 𝑥+ 𝛼 2

𝑒 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑓(𝑇,𝑎,𝑏) ,

+𝑒

𝑦2 1 2 ―(𝑦 ― )( ) 𝜋 𝜀𝑇 𝑦+ 𝛽 2

―(𝑥 ―

+ 2𝑒

𝑥2 𝑦2 1 2 )𝑇2 ― (𝑦 ― )( ) 𝜋 𝜋 𝜀𝑇 𝑥+ 𝛼 𝑦+ 𝛽 2 2

(23)

𝑇,𝑎,𝑏

where ―(𝑥 ―

𝑓(𝑇,𝑎,𝑏) = 𝑒

𝑥2 )𝑇2 𝜋 𝑥+ 𝛼 2

―(𝑦 ―

+𝑒

𝑦2 1 2 )( ) 𝜋 𝜀𝑇 𝑦+ 𝛽 2

(

― 𝑥―

+ 2𝑒

) (

)

𝑥2 𝑦2 1 𝑇2 ― 𝑦 ― 𝜋 𝜋 𝜀𝑇 𝑥+ 𝛼 𝑦+ 𝛽 2 2

2

( )

. (24)

Figure 2 depicts 𝑓(𝑇,𝑎,𝑏) for arbitrary values 𝛼 = 5 and 𝛽 = 10, assuming 𝑎 = 𝑏. It is empirically deduced from figure 2 that 𝑓(𝑇,𝑎,𝑏) is a strictly convex function in observed region, hence it is expected to have only one global minimum.

Figure 2. 𝑓(𝑇,𝑎,𝑏) for 𝛼 = 5 and 𝛽 = 10 (assuming 𝑎 = 𝑏).

According to practical range of parameters in 𝑓(𝑇,𝑎,𝑏), the typical order is 10 ―3 for 𝑇 and 1010~ 1012 for 𝑎 and 𝑏 [16]. Hence it can be easily shown that the last term of 𝑓(𝑇,𝑎,𝑏) can be neglected, compared to first and second terms, which yields

(

― 𝑥―

𝑓(𝑇,𝑎,𝑏)≅𝑒

)

𝑥2 𝑇2 𝜋 𝑥+ 𝛼 2

(

― 𝑦―

+𝑒

)

𝑦2 1 𝜋 𝜀𝑇 𝑦+ 𝛽 2

2

( )

.

(25)

For the sake of simplicity, we approximate 𝑓(𝑇,𝑎,𝑏) using well-known Taylor series expansion 𝑒𝑥 ≈ 𝑥𝑛 𝑛 = 0 𝑛!

∑𝑁

, with 𝑁 = 2. By solving the below system of equations:

6

{

∂𝑓(𝑇,𝑎,𝑏) =0 ∂𝑇 ∂𝑓(𝑇,𝑎,𝑏) = 0, ∂𝑎 ∂𝑓(𝑇,𝑎,𝑏) =0 ∂𝑏

(26)

the optimum parameters (𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) are obtained as: 𝑇𝑜𝑝𝑡 =

( ) 𝛽

1/4

(27)

𝜀2𝛼

𝑎𝑜𝑝𝑡 = 𝑏𝑜𝑝𝑡 = 𝜋

𝛼

(28) 𝛽 The equality of 𝑎𝑜𝑝𝑡 and 𝑏𝑜𝑝𝑡 is noteworthy since it indicates the optimality of matched pulse shaping filters at both the receiver and transmitter sides. Optimum parameters in (27), (28) are the same results derived in [14, 16] with a different approach. The solutions provided by previous studies are based on assumptions which prevent applying additional constraints on Tx and/or Rx pulse shaping filters easily. E.g. the solution provided in [16], assumes < 𝑔, 𝛾 > = const. alongside with normalized 𝑔 and 𝛾 which prevents optimization of Rx or Tx pulse shaping filter when the other is constrained. Also in [14], the parameter optimization problem is solved assuming Rx and Tx pulse shapes to be equal Gaussian pulses. Unlike previous studies, the approach made in this paper generally assumes different Gaussian pulses at Tx and Rx sides, which facilitates applying constraints on Tx and/or Rx pulse shapes, forced by specific applications. Next section provides solutions to pulse and lattice optimization problem with constraints on pulse shaping filters, required by 5G applications, particularly cognitive radio and tactile internet.

4. Optimization for 5G applications Here we apply the developed framework for pulse and lattice scaling to meet the requirements of 5G applications. These applications force constraints on Tx-filter dispersion in frequency and Rx-filter dispersion in time. The temporal and spectral second-order moments of Gaussian pulses given by [14] ∞ 1 𝑡2|𝑔(𝑡)|2𝑑𝑡 = (29) 𝜎2𝑡 = 4𝑏

∫

―∞ ∞

𝜎2𝑓 =

𝑎

∫ 𝑓 |Γ(𝑓)| 𝑑𝑓 = 4𝜋 , 2

2

2

(30)

―∞

represent the time dispersion of Rx-filter and frequency dispersion of Tx-filter, respectively[12], where Γ(𝑓) is the Fourier transform of {γ(t)}. In this section, first we solve the SIR maximization problem with a constraint on the Rx-filter time dispersion in order to have an adequately short CP and as a result, improve the spectral efficiency. Then the same approach is used to limit the Tx-filter frequency dispersion in order to reduce the OOB leakage which is an important issue in 5G cognitive radios. Finally, the SIR maximization is solved considering limits on both Tx and Rx filter dispersion in frequency and time domain, respectively.

4.1. Constrained Rx-filter time dispersion In order to combat ISI in multicarrier transmission, cyclic prefix insertion is performed, accounting for the length of transmitter pulse shaping filter, the receiver pulse shaping filter and the mobile channel impulse response as depicted in Figure 3. Large filter orders are generally problematic due to the cyclic prefix, which should match to the aggregate filter lengths of all system filters involved [26]. However, for the Tx-filter, we can neglect the Tx-filter part by using a tail-biting technique according to [27]. 7

Figure 3. Transmitted block with CP accounting for Tx/Rx filtering and the mobile channel

Satisfying the requirements of tactile internet application in 5G demands high spectral efficiency and ultra-low latency [24]. Limiting the filter length in time improves the spectral efficiency due to limiting the CP length. It also reduces the computational complexity and communications latency [12]. Here we assume the tail-biting technique is used and the CP consists of only channel length and the Rx-filter length. In this regard an Rx filter length constraint as 𝜎2𝑡 < 𝜎2𝑡0 is applied to the SIR maximization problem, where 𝜎2𝑡0 is the maximum limit on Rx-filter time dispersion. Hence we have

(𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑓(𝑇,𝑎,𝑏),

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑏 >

𝑇,𝑎,𝑏

1

(31)

4𝜎2𝑡0 𝛼 𝛽

According to Lagrange method for problems with inequality constraints [28], if 𝑏𝑜𝑝𝑡 = 𝜋

satisfies

1

the constraint 𝑏𝑜𝑝𝑡 > 4𝜎2 , then we already have the solution. Otherwise, the inequality constraint is 𝑡0

1

considered as equality 𝑏𝑜𝑝𝑡 = 4𝜎2 and the problem is solved by the well-known Lagrange method. 𝑡0

Solving the below system of equations:

{

∂𝑓(𝑇,𝑎,𝑏𝑜𝑝𝑡) ∂𝑇 ∂𝑓(𝑇,𝑎,𝑏𝑜𝑝𝑡) ∂𝑎

=0 , =0

(32)

results in the optimum values as below:

𝑇𝑜𝑝𝑡 =

1 4

𝜋𝛽(𝜋𝛼 + 4𝑏𝑜𝑝𝑡)

(

𝛼𝜀2𝑏𝑜𝑝𝑡(𝛽𝑏𝑜𝑝𝑡 + 4𝜋)

)

, 𝑎𝑜𝑝𝑡 =

𝜋𝛼𝑏𝑜𝑝𝑡(𝛽𝑏𝑜𝑝𝑡 + 4𝜋) 𝛽(𝜋𝛼 + 4𝑏𝑜𝑝𝑡)

.

(33)

4.2. Constrained Tx-filter frequency dispersion Overcoming the spectrum scarcity in crowded 5G networks demands cognitive radios to enable flexible aggregation of white spectrum spaces. At the same time, the OOB leakage is desired to be minimized in order to not affect neighboring systems [29]. Similar to the discussion on limiting the Rx filter length in time, it is possible to limit the OOB leakage by applying a constraint on the Txfilter dispersion in the frequency domain. We aim to apply a constraint as 𝜎2𝑓 < 𝜎2𝑓0 to the SIR maximization problem, where 𝜎2𝑓0 is the limit on Tx-filter frequency dispersion. Hence we have

(𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑓(𝑇,𝑎,𝑏) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑎 <

(34)

𝑇,𝑎,𝑏 4𝜋2𝜎2𝑓0

𝛼

2 2 𝛽 satisfies the constraint 𝑎𝑜𝑝𝑡 < 4𝜋 𝜎𝑓0, then we already have the solution. Otherwise, the inequality constraint is considered as equality 𝑎𝑜𝑝𝑡 = 4𝜋2𝜎2𝑓0 and using the Lagrange method [28], the solution is given by:

Similarly, if 𝑎𝑜𝑝𝑡 = 𝜋

𝑇𝑜𝑝𝑡 =

(

𝜋𝛽(𝜋𝛼 + 4𝑎𝑜𝑝𝑡) 2

𝛼𝜀 𝑎𝑜𝑝𝑡(𝛽𝑎𝑜𝑝𝑡 + 4𝜋)

)

1/4

, 𝑏𝑜𝑝𝑡 =

𝜋𝛼𝑎𝑜𝑝𝑡(𝛽𝑎𝑜𝑝𝑡 + 4𝜋) 𝛽(𝜋𝛼 + 4𝑎𝑜𝑝𝑡)

.

4.3. Constrained Rx-filter time dispersion and Tx-filter frequency dispersion 8

(35)

Finally, we solve the problem considering constraints on both Tx filter frequency dispersion and Rx filter length in time. The aim is to maximize the SIR, while 𝜎2𝑡 < 𝜎2𝑡0 and 𝜎2𝑓 < 𝜎2𝑓0 are satisfied. The maximization problem is given by

(𝑇𝑜𝑝𝑡,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑓(𝑇,𝑎,𝑏) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

{

𝑎<

(36)

𝑇,𝑎,𝑏 4𝜋2𝜎2𝑓0

𝑏>

1

4𝜎2𝑡0

Again using the Lagrange method for problems with inequality constraints, we already have the answer if optimum values from equations (27) and (28) satisfy the desired conditions, otherwise, the 1 inequality constraints are considered as equalities 𝑎𝑜𝑝𝑡 = 4𝜋2𝜎2𝑓0 and 𝑏𝑜𝑝𝑡 = 4𝜎2 , and optimum results 𝑡0

are obtained by solving the equation given by

∂𝑓(𝑇,𝑎𝑜𝑝𝑡,𝑏𝑜𝑝𝑡) ∂𝑇

= 0,

(37)

which yields the optimum T as below: 𝑇𝑜𝑝𝑡 =

(

𝜋𝛽(𝜋𝛼𝑎𝑜𝑝𝑡 + 𝜋𝛼𝑏𝑜𝑝𝑡 + 4𝑎𝑜𝑝𝑡𝑏𝑜𝑝𝑡) 𝛼𝜀2𝑎𝑜𝑝𝑡𝑏𝑜𝑝𝑡(𝛽𝑎𝑜𝑝𝑡 + 𝛽𝑏𝑜𝑝𝑡 + 4𝜋)

)

1 4

.

(38)

5. Results and Discussion In this section, the results obtained by optimum pulse and lattice scaling are presented and discussed. We evaluated our method in 5G channels, as described in [30, 31]. Two channel models are defined for link-level evaluations: Clustered Delay Line (CDL) model and Tapped Delay Line (TDL). These models can be scaled in delay so that the model achieves a desired RMS delay spread [30]. Example RMS delay spreads are given in Table I. Table I. Example RMS Delay Spreads for 5G channel models [30] Delay Model Very short delay spread Short delay spread Nominal delay spread Long delay spread Very Long delay spread

RMS Delay Spread (ns) 10 30 100 300 1000

In [31] the following classes of mobility are defined for 5G networks:    

Stationary: 0 km/h Pedestrian: 0 ~ 10 km/h Vehicular: 10 ~ 120 km/h High-speed vehicular: 120 ~ 500 km/h 𝑓𝑐𝑣𝑚𝑎𝑥

The maximum Doppler shift 𝑓𝑑𝑚𝑎𝑥 for these classes is given by 𝑓𝑑𝑚𝑎𝑥 = 3𝑒8 , in which 𝑓𝑐 is the carrier frequency in Hz and 𝑣𝑚𝑎𝑥 is the maximum user velocity in m/s. Assuming 𝑓𝑐 = 2 𝐺𝐻𝑧 , the maximum Doppler shifts for different classes of mobility is obtained as Table II. Table II. Maximum Doppler shift for 5G channel mobility classes [31] Mobility Model Stationary Pedestrian

Maximum Doppler Shift (Hz) 0 18.5 9

222.2 925.9

Vehicular High-speed vehicular

Five scenarios leading to same number of delay-Doppler spread pairs are given in Table III, alongside their corresponding 𝛼 and 𝛽 obtained for 5G channel models. Assuming 𝜀 = 0.93, which corresponds to 6.5% capacity loss caused by normal cyclic prefix in 5G system [31], the optimum values for the different 5G channel models obtained from (27) and (28) are given in Table II. Table III. RMS Delay and Doppler Spread and corresponding 𝛼 and 𝛽 for 5G channel models Channel model short delay pedestrian normal delay pedestrian normal delay vehicular long delay vehicular long delay high speed

RMS Delay spread (ns) 10 100 100 1000 1000

Doppler spread (Hz) 18.5 18.5 222.2 222.2 925.9

𝜶

𝜷

1.5915e+15 1.5915e+13 1.5915e+13 1.5915e+11 1.5915e+11

9.2819e-4 9.2819e-4 6.4458e-6 6.4458e-6 3.7128e-7

Table IV. Optimum pulse and lattice scaling parameters adapted to different 5G channel models Channel model short delay pedestrian normal delay pedestrian normal delay vehicular long delay vehicular long delay high speed

𝑻𝒐𝒑𝒕 0.2866e-4 0.9062e-4 0.2616e-4 0.8272e-4 0.4053e-4

𝒂𝒐𝒑𝒕,𝒃𝒐𝒑𝒕 4.1138e+9 0.4114e+9 4.9365e+9 0.4937e+9 2.0569e+9

The SIR for different channel conditions are shown in Figure 3 with 5G channel models given in Table III. The suboptimum curve is plotted using arbitrary non-optimal values. It is shown that the optimum pulse and lattice scaling improves SIR up to 4 dB. It’s also important to note the optimal SIR values for different channel models in Figure 3, which indicates that using channel-optimum scaled pulse shaping and lattice, the SIR remains almost equal, regardless of channel condition. In the following, the results of optimization with constraints forced by 5G applications are presented considering three scenarios discussed in section 4.

10

12 11 10 9

SIR

8 7 6 5 4 3 2 short(pedest)

optimal SIR a=b=4e6, T=8e-4 a=b=5e6, T=8e-4 a=b=4e6, T=1e-3 a=b=5e6, T=1e-3

normal(pedest)

normal(vehic)

long(vehic)

long(HiSpeed)

channel type Figure 4. Comparing SIR resulted by optimum and non-optimum pulse and lattice scaling in 5G channels

5.1. Results of Constrained Rx-filter time dispersion SIR maximization considering a limit on Rx filter length in time is performed through (34) for different 5G channel models, forcing 𝜎2𝑡 to be 50, 70 and 90 percent smaller compared to its value when no constraint is applied. The obtained SIR is compared to the optimal SIR without the limit on 𝜎2𝑡 , as depicted in Figure 5. It is shown that limiting Rx filter length in time does not degrade the SIR significantly, even if 𝜎2𝑡 is forced to be %90 smaller. In ETU300Hz channel model, which is the worst case, the cost paid in favor of %90 smaller 𝜎2𝑡 , is only 0.02dB degradation in SIR. The reason is that, although the pulse shaping filter in the receiver side, i. e. 𝑔(𝑡), is forced to be narrow in time, 𝛾(𝑡) obtained from (33) will be narrow to match 𝑔(𝑡). Hence although pulses are not optimal due to the constraint, they are still matched together and perform near optimum. 11.52 11.5 11.48

SIR (dB)

11.46 11.44 11.42 11.4

optimal SIR %50 smaller

11.38

%70 smaller %90 smaller

11.36 short (pedestrian)

2 t 2 t 2 t

normal (pedestrian) normal (vehicular)

channel type

11

long (vehicular)

long (High Speed)

Figure 5. Comparison of optimal SIR with/without constraint on Rx-filter length in time in 5G channels

We also investigated the SIR loss due to constraints applied on Rx filter design as depicted in Figure 6. The results show that the SIR loss grows for larger reduction of 𝜎2𝑡 . However it is negligible for all channel models. 10-1 10-2 10-3

Loss (dB)

10-4 10-5 10-6 10-7

short (pedestrian) normal (pedestrian) normal (vehicular) long (vehicular) long (High Speed)

10-8 10-9

0

10

20

30

40 2 t

50

60

70

80

90

reduction percent

Figure 6. SIR loss due to reduction of Rx filter dispersion in time

5.2. Results of Constrained Tx-filter frequency dispersion Similarly, the SIR maximization considering a limit on Tx filter frequency dispersion is performed through (35) for different 5G channel models, forcing 𝜎2𝑓 to be 50, 70 and 90 percent smaller compared to its value when no constraint is applied. The obtained SIR is compared to the optimal SIR without the limit on 𝜎2𝑓, as depicted in Figure 7. Forcing narrow Tx filter frequency dispersion results in more SIR degradation compared to limiting Rx filter length in time, but still, it's not significant. In the worst case, i. e. ETU300Hz channel model, SIR is 0.5 dB degraded due to decreasing 𝜎2𝑓 by %90. Since the scaling rule in (35) retains the Rx filter almost matched to the Tx filter, the performance is near optimal despite constrained pulse shaping at the transmitter side. Figure 8 depicts the SIR loss caused by reduction of 𝜎2𝑓. It can be seen that even appplying high constraint on Tx filter dispersion in frequency leads to less than 1 dB loss in all channel models. Figure 8. shows that the channel with long delay spread and high mobility is most affected by limiting the transmitter OOB. The channel model with short delay spread and low mobility is less sensitive to constraint on Tx filter pulse shape dispersion in frequency.

12

11.6

11.5

SIR (dB)

11.4

11.3

11.2

11.1

optimal SIR %50 smaller

11

%70 smaller %90 smaller

10.9 short(pedest)

2 f 2 f 2 f

normal(pedest)

normal(vehic)

long(vehic)

long(HiSpeed)

channel type

Figure 7. Comparison of optimal SIR with/without constraint on Tx-filter frequency dispersion in 5G channels 100

Loss (dB)

10-2

10-4

10-6

short (pedestrian) normal (pedestrian) normal (vehicular) long (vehicular) long (High Speed)

10-8

10-10

0

10

20

30

40 2 f

50

60

70

80

90

reduction percent

Figure 8. SIR loss due to reduction of Tx filter dispersion in frequency

5.3. Results of Constrained Rx-filter time and Tx-filter frequency dispersion Finally, we evaluate the SIR maximization problem with double constraints on Tx filter frequency dispersion & Rx filter time dispersion and compare it to the same problem without these constraints. Optimization is performed forcing 𝜎2𝑡 and 𝜎2𝑓 to be 30, 40 and 50 percent smaller compared to its value when no constraint is applied. Figure 6. depicts the obtained SIR with and without the limit on 𝜎2𝑡 and 𝜎2𝑓.

13

Unlike previous results for SIR maximization with a constraint on only one of 𝜎2𝑡 or 𝜎2𝑓, limiting both of them leads to more significant SIR degradation. E.g. forcing 𝜎2𝑡 and 𝜎2𝑓 to be %50 smaller, degrades the SIR by 2 dB. The reason is that unlike the previous discussion, here the filter at the transmitter side is forced to be narrow in the frequency domain that results in time domain stretching, while the receiver filter in the opposite is forced to be narrow in the time domain. Hence the Tx and Rx pulse shapes are mismatched which leads to unavoidable significant SIR degradation. 12 11.5 11

SIR (dB)

10.5 10 9.5 9

optimal SIR %30 smaller

8.5

%40 smaller %50 smaller

8 short (pedestrian)

2 t 2 t 2 t

and and and

2 f 2 f 2 f

normal (pedestrian) normal (vehicular)

long (vehicular)

long (High Speed)

channelontype Figure 9. Comparing optimal SIR with/without constraint Tx & Rx-filter dispersion in time and frequency in 5G channels

6. Conclusion The optimization of pulse and lattice regarding channel conditions aiming to maximize the SIR was studied. The optimization was formulated as maximization of SIR by means of scaling Tx and Rx pulse shaping filters, and the T-F lattice, regarding CSI, presented in terms of delay and Doppler spread. SIR maximization problem is then solved analytically, considering the requirements of 5G applications, particularly cognitive radio and tactile internet, which force constraints on Tx filter frequency and Rx filter time dispersions, respectively. It is shown that generally using optimum pulse and lattice scaling, improves the SIR significantly, in comparison to non-optimal pulses and lattices. The results for 5G applications show that applying the constraint to one of Rx filter time dispersion or Tx-filter frequency dispersion does not degrade the SIR significantly while limiting both of them leads to a mismatch between pulse shaping filters and degrades the SIR.

References [1] [2] [3]

S. Tiwari, S. S. Das, and K. K. Bandyopadhyay, "Precoded generalised frequency division multiplexing system to combat inter-carrier interference: performance analysis," Iet Communications, vol. 9, pp. 1829-1841, 2015. S. Ali, A. Ahmad, and A. Khan, "Energy-efficient resource allocation and RRH association in multitier 5G H-CRANs," Transactions on Emerging Telecommunications Technologies, vol. 30, p. e3521, 2019. M. Hawasli and S. A. Çolak, "Toward green 5G heterogeneous small-cell networks: power optimization using load balancing technique," AEU-International Journal of Electronics and Communications, vol. 82, pp. 474-485, 2017. 14

[4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

M. Schellmann, Z. Zhao, H. Lin, P. Siohan, N. Rajatheva, V. Luecken, et al., "FBMC-based air interface for 5G mobile: Challenges and proposed solutions," in Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM), 2014 9th International Conference on, 2014, pp. 102-107. J. Nadal, C. A. Nour, and A. Baghdadi, "Flexible and efficient hardware platform and architectures for waveform design and proof-of-concept in the context of 5G," AEUInternational Journal of Electronics and Communications, vol. 97, pp. 85-93, 2018. E. Dahlman, S. Parkvall, and J. Skold, 4G: LTE/LTE-advanced for mobile broadband: Academic press, 2013. G. Pocovi, K. I. Pedersen, and P. Mogensen, "Joint link adaptation and scheduling for 5G ultra-reliable low-latency communications," IEEE Access, vol. 6, pp. 28912-28922, 2018. H. K. Bizaki, "Towards 5G Wireless Networks-A Physical Layer Perspective," 2016. S. Sandberg, K. Sandlund, and M. Thurfjell, "Radio Link Adaptation In Communication Systems," ed: Google Patents, 2019. S. T. Chung and A. J. Goldsmith, "Degrees of freedom in adaptive modulation: a unified view," IEEE Transactions on Communications, vol. 49, pp. 1561-1571, 2001. T. Strohmer and S. Beaver, "Optimal OFDM design for time-frequency dispersive channels," IEEE Transactions on communications, vol. 51, pp. 1111-1122, 2003. A. Sahin, I. Güvenç, and H. Arslan, "A Survey on Multicarrier Communications: Prototype Filters, Lattice Structures, and Implementation Aspects," IEEE Communications Surveys and Tutorials, vol. 16, pp. 1312-1338, 2014. A. F. Demir, M. Elkourdi, M. Ibrahim, and H. Arslan, "Waveform design for 5G and beyond," arXiv preprint arXiv:1902.05999, 2019. F.-M. Han and X.-D. Zhang, "Wireless multicarrier digital transmission via Weyl-Heisenberg frames over time-frequency dispersive channels," IEEE Transactions on Communications, vol. 57, 2009. B. Le Floch, M. Alard, and C. Berrou, "Coded orthogonal frequency division multiplex [TV broadcasting]," Proceedings of the IEEE, vol. 83, pp. 982-996, 1995. P. Jung and G. Wunder, "The WSSUS pulse design problem in multicarrier transmission," IEEE Transactions on Communications, vol. 55, pp. 1918-1928, 2007. F.-M. Han and X.-D. Zhang, "Hexagonal multicarrier modulation: A robust transmission scheme for time-frequency dispersive channels," IEEE Transactions on Signal Processing, vol. 55, pp. 1955-1961, 2007. J. Du and S. Signell, "Pulse shape adaptivity in OFDM/OQAM systems," in Proceedings of the 2008 International Conference on Advanced Infocomm Technology, 2008, p. 131. G. Matz, H. Bolcskei, and F. Hlawatsch, "Time-frequency foundations of communications: Concepts and tools," IEEE Signal Processing Magazine, vol. 30, pp. 87-96, 2013. W. Kozek and A. F. Molisch, "Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels," IEEE Journal on selected areas in communications, vol. 16, pp. 1579-1589, 1998. A. Sahin and H. Arslan, "Multi-user aware frame structure for OFDMA based system," in Vehicular Technology Conference (VTC Fall), 2012 IEEE, 2012, pp. 1-5. T. Ihalainen, A. Viholainen, T. H. Stitz, M. Renfors, and M. Bellanger, "Filter bank based multimode multiple access scheme for wireless uplink," in Signal Processing Conference, 2009 17th European, 2009, pp. 1354-1358. C.-I. Badoi, N. Prasad, V. Croitoru, and R. Prasad, "5G based on cognitive radio," Wireless Personal Communications, vol. 57, pp. 441-464, 2011. M. Simsek, A. Aijaz, M. Dohler, J. Sachs, and G. Fettweis, "5G-enabled tactile internet," IEEE Journal on Selected Areas in Communications, vol. 34, pp. 460-473, 2016. P. Jung and G. Wunder, "A group-theoretic approach to the WSSUS pulse design problem," in Information Theory, 2005. ISIT 2005. Proceedings. International Symposium on, 2005, pp. 870-874. 15

[26] [27] [28] [29] [30] [31] [32]

G. Fettweis, M. Krondorf, and S. Bittner, "GFDM-generalized frequency division multiplexing," in Vehicular Technology Conference, 2009. VTC Spring 2009. IEEE 69th, 2009, pp. 1-4. H. Ma and J. Wolf, "On tail biting convolutional codes," IEEE Transactions on Communications, vol. 34, pp. 104-111, 1986. P. Davis. (2010, 01/06/2019). Lagrange Multipliers for Inequality Constraints. [course material]. Available: users.wpi.edu/~pwdavis/Courses/MA1024B10/1024_Lagrange_multipliers.pdf M. Matthé, N. Michailow, I. Gaspar, and G. Fettweis, "Influence of pulse shaping on bit error rate performance and out of band radiation of generalized frequency division multiplexing," in Communications Workshops (ICC), 2014 IEEE International Conference on, 2014, pp. 43-48. G. R. A. N. W. Group, "Study on channel model for frequencies from 0.5 to 100 GHz (Release 15)," 3GPP TR 38.9012018. S. Ahmadi, 5G NR: Architecture, Technology, Implementation, and Operation of 3GPP New Radio Standards: Academic Press, 2019. D. A. Swick, "A review of wideband ambiguity functions," NAVAL RESEARCH LAB WASHINGTON DC1969.

Appendix A Here we provide the cross-ambiguity function of Gaussian pulses, which is a slight variation of autoambiguity function of Gaussian pulses derivation, presented in [32]. The cross-ambiguity between 𝑔 (𝑡) and 𝛾(𝑡) is given by [16]: ∞ 𝐴𝑔𝛾(𝜏,𝜈) =< 𝑔,𝑆𝜏,𝜈 𝛾 > = ∫ ―∞𝑔(𝑡)𝛾(𝑡 ― 𝜏)exp ( ―2𝜋𝑖𝜈𝑡)𝑑𝑡 (A-1) Replacing pulse shapes from (17) and (18) in (A-1) we have: 1

4𝑎𝑏 4 𝐴𝑔𝛾(𝜏,𝜈) = ( 2 ) 𝜋

∫

1

∞

4𝑎𝑏 4 2 2 𝑒 ―(𝑎𝑡 + 𝑏(𝑡 ― 𝜏) + 2𝜋𝑖𝜈𝑡)𝑑𝑡 = ( 2 ) 𝜋 ―∞

∫

∞

2

𝑒 ―(𝑎𝑡

+ 𝑏𝑡2 + 𝑏𝜏2 ― 2𝑏𝑡𝜏 + 2𝜋𝑖𝜈𝑡)

𝑑𝑡

―∞

1

4𝑎𝑏 4 =( 2 ) 𝜋

∫

1 4

4𝑎𝑏 =( 2 ) 𝜋

∫

1

∞

2

𝑒 ―((𝑎 + 𝑏)𝑡

+ 𝑏𝜏2 ― 2𝑏𝑡𝜏 + 2𝜋𝑖𝜈𝑡)

𝑑𝑡

―∞ ∞

(

―((𝑎 + 𝑏) 𝑡 ―

𝑒

𝑏 𝜏 𝑎+𝑏

2

2

) + 𝑎𝑎𝑏𝜏+ 𝑏 + 2𝜋𝑖𝜈𝑡)𝑑𝑡

―∞ 𝑎𝑏𝜏2

4𝑎𝑏 4 ― = ( 2 ) 𝑒 𝑎+𝑏 𝜋

∫

∞

(

𝑏 𝜏 𝑎+𝑏

―(𝑎 + 𝑏) 𝑡 ―

𝑒

2

) 𝑒 ―2𝜋𝑖𝜈𝑡𝑑𝑡

―∞

(

―(𝑎 + 𝑏) 𝑡 ―

𝐹.𝑇.{𝑒

𝑏 𝜏 𝑎+𝑏

2

)} (A-2)

Using Fourier transform properties, we have: 2 2 𝐹.𝑇.{𝑒 ―𝜋𝑡 } = 𝑒 ―𝜋𝜈

(A-3) 𝜋2𝜈2

𝐹.𝑇.{𝑒

{

𝐹.𝑇. 𝑒

―(𝑎 + 𝑏)𝑡2

}=

(

𝑏

―(𝑎 + 𝑏) 𝑡 ― 𝑎 + 𝑏𝜏

𝑎 + 𝑏 ― 𝑎+𝑏 𝜋 𝑒

2

)

}=

(A-4) 𝜋2𝜈2 + 2𝜋𝑖𝜈𝑏𝜏 ) 𝑎+𝑏

𝑎 + 𝑏 ―( 𝜋 𝑒

(A-5)

Replacing (A-5) in (A-2) yields the cross-ambiguity of Gaussian pulses: 1

𝐴𝑔𝛾(𝜏,𝜈) =

𝑎𝑏𝜏 2 𝑎𝑏 2 ―( (𝑎 + 𝑏) 𝑒

16

2 + 𝜋2𝜈2 + 2𝜋𝑖𝜈𝑏𝜏 𝑎+𝑏

)

(A-6)

Finally the squared magnitude of cross-ambiguity is given by: 𝑎𝑏𝜏2 + 𝜋2𝜈2 ) 𝑎+𝑏

2 𝑎𝑏

|𝐴𝑔𝛾(𝜏,𝜈)|2 = 𝑎 + 𝑏𝑒 ―2(

(A-7)

.

Appendix B In this part, we provide the derivation of 𝐼(∆𝑇,∆𝐹), i.e. the interference power between symbols with distance ∆T and ∆F in time and frequency, respectively, given by: 𝐼(∆𝑇,∆𝐹) = ∫ℝ2|𝐴𝑔𝛾(𝜏 + ∆𝑇,𝜈 + ∆𝐹)|2𝑆𝐻(𝜏,𝜈)𝑑𝜏𝑑𝜈 (B-1) According to (24) and (25) we have:

𝐼(∆𝑇,∆𝐹) =

𝑎𝑏(𝜏 + ∆𝑇)2 + 𝜋2(𝜈 + ∆𝐹)2 𝑎+𝑏

(

―2 ∬∞ 𝑒 𝑎+𝑏 ―∞

𝑎𝑏𝛼𝛽

)―

𝜋 2

2

(𝛼𝜏2 + 𝛽𝜈2)

2𝑎𝑏(𝜏 + ∆𝑇) 𝜋 𝑎𝑏𝛼𝛽 ∞ ―( 𝑎 + 𝑏 + 2𝛼𝜏2) 𝑑𝜏 𝑎 + 𝑏 ∫ ―∞𝑒 𝑋

𝑑𝜏𝑑𝜈 =

2𝜋2(𝜈 + ∆𝐹)2 𝑎+𝑏

∞ ―( ∫ ―∞𝑒

𝜋

+ 2𝛽𝜈2)

𝑑𝜈

𝑌

(B-2) 2

2𝜋

2𝑎𝑏

Assuming 𝑥 = 𝑎 + 𝑏 , 𝑦 = 𝑎 + 𝑏 we can write: 𝑋 ∞

=

∫𝑒

[(

―∞ ∞

∫𝑒

]𝑑𝜏 =

)

𝜋 ― 𝑥 + 𝛼 𝜏2 + 2𝑥𝜏∆𝑇 + 𝑥∆𝑇2 2

)(

(

)

∫𝑒

[

)

―

]

𝑥2∆𝑇2 + 𝑥∆𝑇2 𝜋 𝑥+ 𝛼 2

𝑑𝜏 = 𝑒

𝑥2∆𝑇2 ― 𝑥∆𝑇2 𝜋 𝑥+ 𝛼 2

―∞

2

𝜋 𝑥∆𝑇 ― 𝑥+ 𝛼 𝜏+ 2 𝜋 𝑥+ 𝛼 2

∞

)(

(

2

𝜋 𝑥∆𝑇 ― 𝑥+ 𝛼 𝜏+ 2 𝜋 𝑥+ 𝛼 2

𝑑𝜏 = 𝑒

𝑥2∆𝑇2 ― 𝑥∆𝑇2 𝜋 ∞ 𝑥+ 𝛼 2

∫𝑒

―∞

(

)

𝜋 ― 𝑥 + 𝛼 𝜏2 2

𝑑𝜏 = 𝑒

𝑥2∆𝑇2 ― 𝑥∆𝑇2 𝜋 𝑥+ 𝛼 2

―∞

𝜋 𝜋 . 𝑥+ 𝛼 2 (B-3)

Similarly, for 𝑌 we have: 𝑌=𝑒

𝑦2∆𝐹2 𝜋 𝑦+ 𝛽 2

― 𝑦∆𝐹2

𝜋 𝜋

𝑦 + 2𝛽

(B-4)

.

Finally, replacing obtained 𝑋 and 𝑌 in (B-2) yields: 𝐼(∆𝑇,∆𝐹) =

𝜋

(

― 𝑥―

𝑥𝑦𝛼𝛽 𝜋

4(𝑥 + 2𝛼)(𝑦 + 2𝛽)

𝑒

17

) (

𝑥2 𝜋 𝑥+ 𝛼 2

∆𝑇2 ― 𝑦 ―

)

𝑦2 𝜋 𝑦+ 𝛽 2

∆𝐹2

.

(B-5)

18