Hybrid Precoding Aided Fast Frequency-Hopping for Millimeter-Wave Communication

The deployment of the Millimeter-Wave (mm-Wave) band in 5G and beyond wireless communications networks is one of the emerging fields owing to its potential of providing extensive bandwidth. Frequency Hopping (FH) has a high potential for use in wireless networks due to its key advantages of spreading the interference over wide bandwidth and of providing diversity gain in counteracting frequency-selective fading. Furthermore, Fast Frequency Hopping (FFH) intrinsically amalgamated with directional Beamforming (BF) may overcome the impairments because of the path-loss of mm-Wave communications. Thus, we propose FFH assisted base-band precoding aided BF for mitigating the mm-wave channel impairments imposed by both fading as well as path loss, whilst relying on a minimal range of radio frequency chains. The mathematical analysis and simulation results demonstrate that hybrid precoded FFH is indeed a promising high-capacity technique of attaining both time- and frequency-domain diversity gains for the mm-Wave communications.


I. INTRODUCTION
The rapid proliferation of mobile devices coupled with increased usage of high bandwidth multimedia application underpins the demand for supporting increased throughput in wireless communication networks [1]- [4]. The recent report of the International Telecommunications Union (ITU) reveals that a minimum peak-rate of 20 Gbps and 10 Gbps for the DownLink (DL) and the UpLink (UL) is required accordingly [5]. The limited spectral resources, available in the lower-part of the current microwave band, fail to meet this ambitious data rate requirement [6]- [9]. Nevertheless, high frequencies transmissions can support substantially higher bandwidth and are currently under utilized [10]. Therefore, the millimeter-wave (mm-Wave) band appears to be a potential solution. It has hence attracted substantial research interests due to benefit of wide bandwidth available [11], [12].
The associate editor coordinating the review of this manuscript and approving it for publication was Wenjie Feng. However, the key challenge is that mm-Wave frequencies are severely affected both by a high path loss and by a frequency-selective as well as time-selective fading [13]- [15].
The impediment of severe path loss may be mitigated by employing a highly directional array aided Beam Forming (BF) schemes [16]- [18]. However, as these BF schemes require accurate channel knowledge which is difficult to acquire due to operation over large antenna arrays, gigasample mixed signal devices or wider bandwidths, therefore, new and low complexity signal processing techniques are required [10], [17], [19]. Channel estimation approach has been adopted for mm-Wave systems in [18], [20], [21] and references within. While to avoid gigasample mixed signal devices have been adopted in [3], [22], [23], and references within. However, in this manuscript we conceive a very unique methodology for hybrid digital to analog (DA) precoders by employing Fast Frequency Hopping (FFH) aided BF schemes.
Frequency diversity schemes rely on generating multiple frequencies to address these prevalent challenges for transmitting the intended data streams [24], [25]. Therefore, they are capable of mitigating both the fading as well as the interference because the extended periods of deep cochannel fading or high interference, are avoided by hopping to an independently fading and/or interfered frequency [24], [26], [27]. Indeed, when considering FFH systems, during the transmission of a specific data symbol multiple hops maybe traversed [24]. Furthermore, FFH models are also capable of attaining improved gains as a benefit of randomizing bursty errors [24], [28]- [30]. Additionally, FFH models are capable of supporting a huge range of users [31]- [33]. In some of the FFH systems the identical symbols are repeated during several hops, which are associated with a different frequency and a different time slot. The various replicas of the signal are more likely to be subjected to independent fading conditions, especially if the channel fading is fast and can be considered uncorrelated from symbol to symbol. This makes it highly likely that one of the hops, experiences reduced fading [24], [29]. Moreover, the signals received from different hops can be combined for reducing the channel effect significantly at the receiver [24] to reduce the impact of prorogation impairments at the receiver. In other words, diversity combining can improve Signal-to-Interferenceplus-Noise-Ratio (SINR) at the receiver side [34]- [38]. One of the earliest notable contributions on FFH diversity combining was made by Lee et al. [34]- [38], which was followed by Wang and Jiang [39]. As a further development Teh et al. [40] have investigated self-combining based Binary Frequency Shift Keying (BFSK) receivers in Rician fading environments.
Directional BF may be used for mitigating the propagation losses in mm-Wave systems [41], [42], which have substantial bandwidth reserves. However, numerous antennas have to be adopted for achieving a high BF gain by relying on λ/2 spacing between the antennas, where λ is the wave length [8], [43], [44]. Fortunately, the wavelength is as short as 0.5 cm at 60 GHz, hence numerous 0.25 cm-spaced antennas may be accommodated even by a shirt-pocket-sized handset. However, if a fully digital transmit precoder scheme is adopted in mm-Wave systems, the RF chains will consume excessive power and the hardware complexity will also increase dramatically [17], [45]. By contrast, analog precoders are less complex than digital precoders. Hence, the concept of hybrid BF, where digital processing was ingeniously combined with analog BF [6], [17], [46]- [48]. Hence, a hybrid Digital-to-Analog (D-to-A) precoding model which incorporated with a range of antennas and a small range of RF chains constitutes a compelling technique of balancing the power consumption, the implementation complexity and the performance of these systems. In hybrid systems, analog BF is implemented by the analog phase-shifters with the RF stage, after the digitally precoded signal is forwarded to the transmitted antennas. At the RF end, the RF signals are first phase-shifted and they are processed by the digital combiner in the base-band system.
Currently, in mm-Wave technology two types of methodologies are opted for the hybrid (D-to-A) precoding. Namely first is referred to as Full-Antenna Array Structure (FAAS) and while other a Sub-Antenna Array Structure (SAAS). In the FAAS model, all the transmit antennas are coupled with dedicated RF chains. In this approach the FAAS achieves full analog BF gain, as in this case the number of antennas is greater than the number of RF chains whereas in SAAS, the computational complexity is compacted at the cost of reduced BF gains [23], [49]. It may be anticipated that the later releases of the 5G cellular network will have a mm-Wave band mode [50], [51]. The 4G UL system depends on Single-Carrier-Frequency-Division Multiple-Access (SC-FDMA) [52], which may also find its way into Next Generation (NG) systems. It is also eminently suitable for FH [14], [24].
The Frequency Division Duplex (FDD) models, the DL is calculated at the receiver, then is quantized and feed back to the Transmit Precoder (TPC), which in turn exploits the knowledge of the Channel Impulse Response (CIR) for pre-compensating its effect. More explicitly, based on its CIR estimates, the best TPC matrix is selected from the code-book and feeds back its index to the transmitter. This procedure is often termed as Limited-Feedback Design (LFD). In this context, Song et.al designed a code-book for mm-Wave systems utilizing the Orthogonal Matching Pursuit (OMP) technique [53], [54]. A finite modulation alphabet is was designed by Rajashekar and Hanzo [55], which relies on a vector quantization based formulation. Alkhateeb et-al [17] designed a code-book for mm-Wave systems using RF features. Furthermore, Clerckx et-al [56] introduced the performance analysis of the code-book design, which works according to the codebook entries. In the light of the above discussion, we conceived a low-complexity hybrid beam forming code-book design. Against this background our novel contribution are as under: • We propose a novel implementation of FFH system in mm-Wave channel by using linear combination techniques. We also compare two linear combination techniques to see which scheme works better in mm-Wave channel.
• We propose FFH aided digital BF assisted with Discrete Fourier Transform (DFT) supported analog RF BF for a limited-feedback assisted mm-Wave system.
• Lastly, we analyse the outage capacity of the resultant system in mm-Wave channels.
Our system model is illustrated in Section II. In Section III, we describe the mm-Wave channel, while in Section V-A, we portray the DFT aided code-book model invoked for analog RF BF. In Section VII, we analyse simulation results followed by conclusions in section VIII. Before proceeding to our system model, Symbol generated by k th user.

II. SYSTEM MODEL
The architecture of the system proposed is depicted in Fig. 1, which comprises of three parts: the transmitter, the mm-Wave channel and the receiver in detailed in the succeeding sub-sections.

A. TRANSMITTER MODEL
A basic architecture of FFH-Multiple Frequency-Shift Keying (MFSK) transmitter is shown in Fig. 2. It is anticipated that all communication from mobile User Equipment (UE) is scheduled by the BS on the link which employs FFH-MFSK transmission [24], [57]- [59]. It is reflected from Fig. 2 that the enormous data rates consists of sub-stream U , where the operation of MFSK and FFH are carried out at each sub-streams to transmit the data to the receiver. From Fig. 2, the period of the binary input is T b having a rate R b of the first Serial-to-Parallel (S/P) converter to transfer U parallel sub-streams. In the process of symbol-duration T s , every sub-stream generates b = log 2 (M ) bits which form a MFSK symbol [22], [34], [60]. In the attempt to execute the FFH, the duration of the symbol is denoted as T s = UbT b . Additionally, the T s is divided into L chips or hops where T h can be quantified as T h = T s /L, and represents the number of time-slots duration. In FFH model illustrated in Fig. 2, where the system bandwidth is shared by the total number of K users. The k th user address is randomly and uniquely assigned by utilizing the Pseudorandom-Noise (PN) code and is represented as where l = 0, 1, · · · , L − 1, a k (l) ∈ GF(M ) and GF(M ) represents the Galois Field with M number of elements. X k is the first designated symbol and Thus, all-one vector of length L is denoted by 1 L and GF of addition is represented as ⊕ which operates as the additionmodulus. After a carrier frequency f c is modulated by the MFSK modulator, the modulation signal is broadcasted over the channel. Specifically, let us suppose the transmitted signal k-th user is denoted as where the transmitted power signal is P s , the carrier frequency is f c , the frequency tone is f (k) l which is calculated by utilizing the information of X k and a k (l), and the normalised pulse waveform is p(t) having duration T h . φ k l is the initialization phase for the MFSK. Now all the transmitted signals will be Inverse Fast Fourier Transformed (IFFT) and stored as a stream s s s vector, where K ≤ N s . The generator/transmitter side is associated with N RF t antenna chains and the RF beam-former matrix is F F F RF of size

III. mm-WAVE CHANNEL
The mm-Wave channel is employed with the widely used of multi-path system, in which a number of distribution paths are linked with a little scatters [4], [9], [19], [57]. The statistical channel model H H H response is represented as n c is the approximation to the complex-valued Gaussian random variable denoted by ≈ CN (0, 1), which has uniform distribution containing Rayleigh phase and amplitude. The response vector a r and a t of Uniform Linear Array (ULA) having antenna elements N r and N t are represented as:

IV. RECEIVED SIGNAL
Now considering a single mm-Wave model depicted in Fig. 1, where the receiver is provided with N r receive antennas. After the RF and base-band combination the received signal vector is represented as where the receiver side is linked with N RF r chains, the RF combiner matrix size is N RF r × N r denoted by W W W RF , the base-band combiner having size N s × N RF r denoted by W W W BB , respectively. The received vector is y y y of size N s and the noise vector n of independent and identically distributed CN (0, σ 2 ).
In Fig. 1 the mm-Wave based architecture is presented, where F RF and W RF are expressed as analogue BF. The beams are phase shifted, which are navigated and guided in the direction of the desired users. The base-band precoder are F BB and combiner weights are W BB , which are utilized for proliferating the inter-stream interference as indicated in

V. CODE-BOOK DESIGN
The TPC and analog BF transceiver design is shown in Figs. 3 and 4. In this design, the information data are precoded prior to the transmission. The base-band utilizing a linear TPC, then the RF beamformer precoded data are transferred. The data are transferred to the transmitted antennas through a phase shift procedure. The RF beamforming are assisted by the DFT, whereas antenna weights are chosen for obtaining desire navigation in the suitable aspect.

A. SUB-ARRAY DESIGN (SAD) CONNECTED ARCHITECTURES
In this design the base-band signal is precoded utilizing a linear TPC approach. The sub-arrays are separated by the full-connected array design which can be seen in Fig. 5. In the Sub-Array Design (SAD) approach, the sub-arrays are split according to the distance of λ/2 and individual RF chain is coupled with the sub-array for digital processing as illustrated in Fig. 5, so that the channel becomes geometrically correlated across all the antennas. This condition depends only on the Angle of Departure (AoD) of the signal at the transmitter, which is calculated in the Eq. 6 of transmitter antenna a T t . More importantly, the a T t is related to the function of AoD which is same when the full rank is greater than the channel matrix H thus making the channel correlated. This condition depends upon two factors, namely the distance between two sub-arrays should be less and the minimum separation distance should be λ/2 and other is the angular spread of the AoD. This is measured from one sub-array to another and should be small which effects the correlation of the given signal. Presuming that the sub-array i of the transmitter is transmitted through the H i then the channel matrix is expressed as Please note that in full-array-connected design, the number of phase shifters needed is always greater than the SAD approach. Assuming that the transmit antennas N t are connected to the RF chain N RF t so the full connected design requires N t × N RF t phase shifters but the SAD would require only N t phase shifters.

B. RF ANALOG BEAMFORMER DESIGN USING DFT
In mm-Wave the propagation losses are due to the foliage, oxygen absorption and water vapour thus by adopting BF we can reduce them. In BF case the antennas are separated by λ/2 distance [6], [8], [23], [61]. We explore the optimal beamforming weight vectors of the statistical distribution matches identically to the DFT based code-book for RF beamforming, making the correlated channel more effective and efficient in terms of DFT based code-book [44]. Moreover, the constant modulus helps in communication system to avoid power imbalanced, which is introduced by the phase shifters [24], [32]. Thus, utilizing the analog RF beamforming using phase shifters is sensible approach in DFT based code-book [24], [32], [44]. Beside, for the base band TPC matrix, we design a code-book which works on the principle of mutually unbiased bases (MUB)s to achieve maximum efficiency despite of having low complexity. The analog RF beamformer F RF having a column of N RF t is obtained from the DFT matrix having size of N t × N t , which reveals that the singular matrix of the channel H = U V H has the maximum correlation at the extreme right side of the channel that is V and RF beamformer is expressed as where the i th column is DFT N t (:, i) having DFT matrix N t × N t , the right singular matrix of the channel is V . The column of W RF are N RF r which are selected from the DFT matrix of order N r × N r , this DFT column generates the maximum correlation of the channel H located at the left hand side of the singular matrix where the i th column is DFT N r (:, i) having the N r × N r DFT matrix, the left singular matrix of the channel is U . Therefore, on the channel analog RF beamformer weights are applied by the base band TPC, the effective channel can be expressed as

C. BASE-BAND CODE-BOOK DESIGN MUBs
The methodology of code-book design and MUB is explained in this passage. Let a set of orthonormal base of dimension N of Hilbert space are C and C . According to the information given in [62], where C and C are mutually unbiased if and only if this condition exists.
Here c, c are vectors belong to C and C respectively. Thus, the number of MUBs is restricted in a given dimension, so that the code book build from MUBs should have a minimum distance as expressed in [44]. It is mandatory that the number of antennas should be in prime power as illustrated in [56], which leads to the information that the number of MUBs antennas expressed as N RF t having a dimension of N RF t + 1. By doing this the values of the MUB matrix will be of finite alphabets which are 1, −1, i, −i. Moreover, if F 1 , · · · , F N are codewords then the distance between the two successive numbers should be minimum expressed in [44], [62] and given by

D. CODE BOOK CONSTRUCTION
For a code-book construction, we use a Hadamard Matrix D of dimension size 2 n where a natural number is n, which obeys the general condition of a MUB. Thus, utilizing the concept of Hadamard matrix D, we are generating a matrix E having same matrix size that of the D matrix. Now let us find the column vector v, by estimating/calculating the elements present in C H v . 1 Now considering the concerned matrix D, where each element is obtained by element-wise multiplication of v with C where C = c 1 , c 2 , . . . , c 2 n so our concern matrix D = v ⊗ C 2 n = [vc 1 , vc 2 , . . . , vc 2 n ]. This process will be continued until all the 2 n + 1 values are obtained for the calculations. Now let us consider the system where we have four transmitted antennas so the RF becomes 1 where the magnitude is equal to 1.
The above equation containing 1 2 satisfies the basic concept of [55]. The vector v 1 is selected as the magnitude of all the elements becomes equal to 1 when we achieve all the calculation of A H v i . For the solution, we assume following three possibilities Now solving the above equation we get the set of MUBs which are given by Since the results obtained from the above Eqs. illustrates that we have identical matrix which satisfies all the condition of [55]. Therefore, obtaining the MUBs N RF t + 1 = 5, the input streams are non-linear, hence the linear process cannot occur in the identity matrix, so we need to dumped the feedback overhead. The left circular shift of each matrix will obtain the code-book entries for the concern matrices C, D, E, F as we are doing 4 shifts so C = 2 4 = 16 entries will be there. Therefore, the set of TPC matrices will have following entries for the code book After obtaining the code-book C A , we select and obtain the first column of the specific TPC matrix having the minimum SNR of all the links of transceiver design of the concerned channel where the concerned noise power is expressed as N 0 and the signal power is P respectively. Therefore, the TPC F BB which encounters the minimum SNR is selected by the code-book C A which obeys where the minimum singular value is min of H eff F BB . Moreover, the mm-Wave channel is correlated when λ/2 is the space between two adjacent antenna elements. Therefore using MUB based TPC structure with DFT assisted RF BF minimizes the intricacy, thus improving the performance in spatially correlated channel. The idea of our design pseudo code is listed in Algorithm 1: The complexity in the code-book is related to the number of estimations needed for achieving and obtaining the best TPC matrix. For achieving the TPC matrix from the code-book we need to calculate and accomplish CN RF r (N RF t − 1) which is complex valued addition as expressed in [60], [63], the cardinality of the code-book is C. Note that its complex value addition and it cannot treated as complex multiplication. Where as, when the Grassmannian and Fourier design are used, we need to perform complex multiplication plus addition. Thus, we obtained CN s (N RF r ) 2 plus C(N RF r ) 2 (N s − 1) as illustrated by [61]. Moreover, using TPC in the base-band we operates on perfect CSI, such that utilizing similar number of RF chains and only if we utilized to use complex value N s N RF r (N RF t − 1) addition plus the multiplication of N s N RF r (N RF t − 1) which shows that the MUBs based code-book design reduces overall complexity when using it in the BF structure. Hence, the values of codebook are made from a finite alphabets = i, −i, 1, −1, which has constant modulus constraints and thus helps in providing the power imbalance to the system. Please note that the computation performed by the transmission of the higher rank are associated with the code-book, which is reused for the transmission of lower rank systems, as they are subset of the higher rank transmission.

VI. FFH NON-COHERENT MFSK RECEIVER
We examine the behaviour of an FFH aided non-coherent MFSK receiver utilizing linear combining scheme for perceiving the generated signal as shown in Fig. 1. If the generated signal is mitigated by the noise and passes through the interference-free channel, then during the l th hop duration T h the signal received is represented as The channel gain is expressed as h (k) l which corresponds to the k th user for the frequency as f (k) l and the complex Additive White Gaussian Noise (AWGN) is represented as N (t) having zero mean. The noise power spread over the complete spread spectrum bandwidth of R is N 0 times B. Therefore, the signal power P to the noise power ratio over the full hopping bandwidth R is expressed as where the energy per transmitted symbol is E s , the transmitted symbol duration is T s = LT h and FSK signal bandwidth is denoted by R = 1 T h . The signal noise power ratio over a single frequency band of M-MFSK tones is given by E s /LN 0 and than corresponding to a single FSK signal having a bandwidth is B may be expressed as E s /LN 0 .
In Fig. 5 the MFSK detector preventing the decision making as shown having non-coherent detection. It is visualized that after some normalisation the samples of input can be expressed as , m] + n ml | 2 m = 0, 1, · · · , P − 1; l = 0, 1, · · · , L − 1 (22) where Dirac Delta function is δ[x, y] which exist as δ[x, x] = 1 and δ[x, y] = 0 for x = y, complex Gaussian noise sample distributed with zero mean is n ml and a variance σ 2 = LN 0 /E s , where the energy per MFSK symbol is denoted as E s = PT s . In Fig. 5, the non-coherent detector is either a single user detector or multi user detector, which operates on the hard decision criteria as done in [24], [64]. Referring to Fig. 5, where linear combination model is used to integrate all the hops available in the signal. The decision variables are computed by the diversity combiner output M . The decision is gathered on the principle of highest magnitude value referred as Maximum Likelihood (ML). The decision index variable generates the estimation of the transmitted symbols. A decision error can occur only if the decision amplitude corresponds to the non-signal tones, which refers to the one tone which exceeds the transmitted signal tone.
where the AWGN components are N 0 = N 2 0c + N 2 0s respectively. The output of the detector associates with the generated signal tone as expressed in Eq. (23) and the remaining (P − 1) detectors associate with non-signal tones, it is envisioned that thermal noise can mitigate its output by performing superposition principle. Therefore it is expressed as The above Eq. (24) reveals that the output is mitigated by noise and the interference present in the signal which effects all the tones M present in the band.
Accordingly, the noise mitigates the signal and it significantly deteriorates the signal. In the scenario of FFH models, all the hops experience noise and jamming independently whereas jamming is low to overcome this linear combination scheme is utilized.

1) TRANSMITTER SIDE
The generated signal of the k th user is represented as l ) (27) Here, the transmitted power is expressed as 'P', the rectangular pulse-shaped signalling waveform is expressed as P Th(t) which is linked with a chip time interval as the phase φ (k) l introduced by MFSK.

2) RECEIVER SIDE
The signal s(t) is modulated by BFSK over a mm-Wave fading channel which is expressed at the receiver side by where the Rayleigh distribution is expressed as α as shown in Eq. 28. The channel-induced phase shift is expressed as φ in the generated signal and the Gaussian Distribution Noise is denoted by n(t). The first symbol interval is generated by the l th tone. The received signal without noise is denoted as Now combining the generated signal by the k th user's without the noise is denoted as l ) (30) VOLUME 9, 2021 FIGURE 6. Outage capacity p out = 1% for Sub-array-connected system for N t = 128, 1024 antennas with and without FFH system.
where the time delay is τ k for the k th user. Let us suppose a signal is corrupted by a noise in the system, thus making it difficult to predict the unwanted signal because of its random behaviour [24], [65]. It is often narrated as that noise is one of the major factor which effect the signal directly [24], [65], so in other words the generated signal are affected by noise independently. Therefore, the received signal are expressed as (31) where, the AWGN is expressed as N(t). It can be noted that the receiver for FFH signal is inverse structure of the transmitter of Fig. 2. The FH pattern is utilized to perform down conversion for the received signal which is multiplied by the carrier frequency as illustrated in Fig. 6. Referring to Fig. 6, the signal is first transferred to a bank of energy detector, where the MFSK procedure is used to convert the obtained signal which is referred to down converted signal. The signal received, uses the identical information as that of the transmitted signal. Please note that the FH intervals of the desired user are synchronized with the detection interval FH time as expressed by T h . The energy detectors of M produce a output of M × L, where L chips are associated with M -ary symbol interval of T s second. In Fig. 2 at the transmitter side MFSK signal converts a 'b' bit symbol to the unique sequence which is called as user address. The output Y k is received at the receiver, where as energy detection is performed to obtain the generated symbol of X k belonging to k th user, where modulo two is carried out to generate the output. The purpose of this is to subtract the user address a k from the Y k information. Thus, leading to the equation.
where the vector of the length L have zero value. The time frequency matrices dimension is M × L where rows are M and columns are L, which describes the detection of the user address at the receiver. The MFSK frequency stages has M rows having bandwidth R h and L narrates the corresponding chips having time duration T h and symbol time interval T s . The identical user-address a k = [5,2,6,4] is used at the receiver out Y k = [0, 5,1,7] where as the same user-address is used for de-hopping.

VII. SIMULATION RESULTS
The performance of our FFH in mm-Wave channels is evaluated in this section. Our analysis compares the performance results with and without FFH for mm-Wave communication.
Before evaluating the Probability of Error P e , we quantify the outage capacity, using DFT-MUB matrix in mm-Wave system by considering operation in both the FFH and non-FFH mode. According to Shannon Capacity, the system works well with finite error probability by utilizing finite blocks for the estimation of the system. By contrast if the infinite block length are used then the outage capacity is calculated which achieves better performance as compared with the finite block lengths of Shannon work [60], [66], [67]. Therefore, to visualize the system, we calculated the outage capacity.

A. OUTAGE CAPACITY
The model is said to be in outage condition only if, when at a given threshold rate the data rate falls below it as illustrated by [66], [67]. The Shannon capacity is measured in [46], which fall below the threshold rate of C out . Moreover, the outage probability p out is calculated by the outage capacity as where the probability is denoted by P, the Rayleigh distribution of fading is h and the sum of squares of random variables having the distribution of Chi-square is given by ||h|| 2 with degree of freedom by 2N sub , where N sub is the number of sub-arrays present in the system. Accordingly [66] the above Eq. (33) becomes as where SNR = N t P N 2 sub N 0 , N t N sub deals with the beam forming while the power allocation for all the sub-branches is 1 N sub . Fig. 6 depicts an outage capacity versus given SNR, when the outage probability is fixed to p out = 1%. In this figure, FFH is employed using two combining schemes such as the Equal Gain Combiner (EGC) and the Selective gain Combiner (SC). We have also compared the performance of the mm-Wave system when FFH is not employed. We have also incorporated different number of transmit antennas such as N t = 128 and 1024. It can be observed from the figure that as the number of transmit antennas increases the performance becomes better because of the transmit diversity gain. Furthermore, the EGC scheme outclass the SC scheme for a given SNR. This improvement is because EGC employs co-phases the signals of each branch and then combines them with equal weighting. While the SC only combines the output branch which has the highest SNR. From the result of Fig. 6, it can be deduced that FFH can be employed for sub-array architecture for mm-Wave system.

B. ACHIEVABLE RATE
We analyse the achievable rate while using the Sub-connected Array using FFH system in mm-Wave model. The Monte Carlo simulation are performed for evaluating the achievable rates of linear combination scheme of EGC utilizing SVD, SC utilizing SVD, unconstrained digital precoding utilizing SVD, Orthogonal Matching Pursuit (OMP), DFT-MUB codebook and DFT-Identity, for 8 × 8 and 32 × 16 MIMO systems. For DFT-Identity precoder, the identity matrix is selected as the baseband TPC are depicted in Fig. 7. 4-bit feedback is evaluated for the DFT-MUB based codebook model. From Fig. 7   DFT-MUB for 8 × 8 MIMO matches with the OMP. The comparison between all these schemes are carried out in mm-Wave channel using DFT-Identity aided model, here the TPC is formulated by choosing first N s columns of the identity matrix. Approximately 0.72 dB gain is calculated while using DFT-MUB with DFT-Identity aided model. Identical performance is obtained for the DFT-Identity and DFT-MUB, as the channel is correlated, beside the performance gain of the MUB code-book is marginally correlated in the base-band channels. Fig. 8, depicts the BER performance results of SC, when deployed in FFH system using mm-Wave channel. We evaluated the performance gap between the SC through Monte Carlo simulation. In this configuration, received antennas can be varied while keeping transmit antennas fixed. A 3 dB gain is obtained by varying receiving antennas from 2− > 8. We see that SVD of each system behave better when employed in FAAS system. It is observed that adopting FFH schemes in mm-Wave system with employing the DFT-MUB code book. The FFH behave better, when N r = 8 we achieve 3 dB gain in a particular SNR and when system parameters are same we achieve almost 1 dB we have one entry for detecting M × L entries. Fig. 8, depicts that by using FFH system, we can achieve better gains despite of little system complexity. Fig. 9 demonstrates that the performance of EGC when deployed in mm-Wave channel. In Fig. 9 when we employ more antennas we achieve a better gain. In Fig. 9, transmitter antennas are fixed and receiving attennas varies. We achieve better gain as soon as more antennas are deployed at the receiver. In this respect, we attain 3 dB gain when N RF t are increased from 2 to 8. This result indicates that diversity is gained because of two factors namely, the antennas increased at the receiver and other linear combination schemes. Fig. 10 shows that the EGC attains better BER performance versus the other schemes used in mm-wave channel. Furthermore, it can be observed that FFH EGC-SVD at 10 −4 BER achieves 6 dB where as FFH SC-SVD achieves 7 dB, unconstrained SVD mm-Wave achieves 8 dB and DFT-MUB code-book is around 12 dB which indicates that the FFH EGC-SVD performs better and obtain better gain because of diversity gain as at lower SNR FFH-SVD SC and SVD of DFT-MUB code book behave better. Moreover, with BER at 10 −4 the performance gap between unconstrained SVD, SVD of SC and SVD of EGC become more because at higher SNR values we get better maximum values of U m has at these values of SNR we achieves optimum number of entries to eliminate from individual rows in the detection matrix, which results in best performance than EGC.

VIII. CONCLUSION
A hybrid code-book model is evaluated with the aid of the MUB base-band and RF in the DFT. The FFH model is utilized for mm-Wave channel. We demonstrated that the proposed design exhibits progressive performance in the form of achievable rate and moreover, we achieve better gains which is within 2 to 3 dB from the SVD precoding. The scheme imposes significantly low complexity than OMP and SVD. A novel design is proposed for the FFH model, which caters for mitigating the fading environment for mm-Wave channels. Moreover, simulations demonstrate that by utilizing FFH model we attain better gain than the existing schemes in the mm-Wave channel. It was demonstrated that the FFH system provides extra diversity gain when employed in BF. Finally, the DFT-MUB code-book is obtained on the contrary of the FFH system model in order to estimate how much gain is achieved in-terms of achievable rate and outage capacity. It was observed that by adopting linear combination schemes in FFH system, we achieve better gain and better outage capacity then that of conventional mm-Wave system. EGC performs better than SC combiner when used in FFH system.