Modulation Symbol Pulse Shaping Transceiver for Affine Frequency Division Multiplexing

1 Modulation Symbol Pulse Shaping T ranscei ver for Af fine Frequency Di vision Multiplexing Haojian Zhang, Graduate Student Member , IEEE , Jiayan Y ang, T ingting Zhang, Member , IEEE , Xu Zhu, Senior Member , IEEE Abstract —The recently pr oposed affine frequency di vision multiplexing (AFDM) wav eform can adjust the time-frequency diversity gain by tuning chirp-rate parameter . Therefor e, it is a candidate wa veform in doubly-selective channels. This letter re veals that the modulation-symbol-domain shaping pulse of AFDM is generated by a con volution-like operation between the time-domain and frequency-domain shaping pulses, indicating that the modulation-symbol-domain pulse shaping of AFDM can be achieved by separately shaping in the time domain and frequency domain. Based on this, this letter presents an AFDM modulation-symbol-domain pulse shaping transceiver which has an ability to achieve the Nyquist pulse shaping, and pro vides the corresponding input-output relationship. Numerical results demonstrate the effectiveness of the proposed transceiver in impro ving the channel sparsity and pilot-to-data interference. Index T erms —Affine fr equency division multiplexing, pulse shaping, doubly-selective channels, wireless communications. I . I N T R O D U C T I O N I N recent years, various modulation waveforms have been proposed for the wireless communications in doubly- selectiv e channels [1]-[5]. Among these wav eforms, the chirp- based affine frequency division multiplexing (AFDM) can adjust the modulation symbol dispersions in doubly-selectiv e channel through using channel-customized chirp-rate parame- ter , thereby obtaining time-frequency diversity gain [5]-[7]. AFDM has become a candidate wa veform under doubly- selectiv e channels, and multiple studies hav e been conducted, such as channel estimation, channel equalization, multiple input multiple output, and multiple access [8]-[14]. Pulse shaping is an important step in digital signal process- ing, which adjusts the tail attenuation features of discrete sam- ples in the continuous domain at the cost of potentially increas- ing the bandwidth of the dual domain by Fourier transform. For instance, time-domain pulse shaping can rapidly attenuate digital signals in the time domain, thereby reducing the time- domain guard interv al between the two adjacent signals for av oiding inter-symbol interferecnce. For wireless communica- tion signals, pulse shaping in the modulation symbol domain also plays an important role, as it determines the sparsity of the modulation-symbol-domain channel, and the interfer - ence between modulation symbols after passing through the The source codes for the simulations in this paper will be released on GitHub . The corresponding URL will be provided in the subsequent version of manuscript. The authors are with the School of Information Science and T echnology , Harbin Institute of T echnology (Shenzhen), Shenzhen, 518055, China. (e-mail: zhanghaojian@hotmail.com; jiayanyoung@gmail.com; zhangtt@hit.edu.cn; xuzhu@ieee.org) channel. The research in [14] rev eals the significance of inter- pulse interference, which influences the accuracy of channel estimation, and inter-region interference, which characterizes the pilot-to-data interference, in the embedded-pilot-based AFDM wav eform. Moreover , the research in [14] verifies that modulation-symbol-domain pulse shaping can improv e the performance of AFDM via direct time-domain transmission and reception windowing. Ho we ver , the analysis of the pulse shaping mechanism in [14] is still insufficient, as it does not clearly indicate the coupling from time-domain shaping pulses in the input-output relationship of AFDM. Moreover , the proposed direct time-domain windowing method cannot achiev e Nyquist pulse shaping, which remains to be improved in shaping fle xibility . By conducting a further analysis of the input-output rela- tionship, this letter reveals that the modulation-symbol-domain (also known as the discrete affine Fourier transform domain) shaping pulse of AFDM is generated by a con volution-lik e operation between the time-domain and frequency-domain shaping pulses, indicating that the modulation-symbol-domain pulse shaping of AFDM can be achie ved by separately shaping in the time domain and frequency domain. Based on this, this letter presents an AFDM modulation-symbol-domain pulse shaping transceiver and provides the corresponding input- output relationship. The proposed method can adjust the pulse width of the modulation symbol domain by adjusting the c yclic prefix and postfix, impro ving the flexibility of AFDM pulse shaping. Numerical results demonstrate the effecti veness of the proposed transceiv er in improving the channel sparsity and pilot-to-data interference. I I . P U L S E S H A P I N G A N A L Y S I S I N A F D M This section provides the deri vation of input-output rela- tionship and the deriv ativ e pulse shaping analysis in AFDM wa veform. A. T ime-domain signal model The receipt baseband signal can be express as s R ( t ) = g t, R ( t ) ∗ P − 1 X p =0 α p s T ( t − τ p ( t )) e − j 2 πf c τ p ( t ) ! (1) where ∗ denotes the linear con volution operator . α p ∈ C and τ p ∈ R denote the path gain and time-varying delay of the p -th path in propagation channel, respectively . g t, R ( t ) denotes the 2  󰆒        󰆒     󰆒 = 0  󰆒 = 1  󰆒 = 2  󰆒 = 3  󰆒 = 4  󰆒 = 5   󰆒          󰆒      󰆒     󰆒          󰆒          󰆒          󰆒          󰆒            󰆒      󰆒          󰆒          󰆒          󰆒          󰆒             ,  󰆓 ,    = 1 3  = 1 2  = 1 1  = 1 0  = 9  = 8  = 7  = 6  = 5  = 4  = 3  = 2  = 1  = 0 Fig. 1. Illustration of the pulse shaping in AFDM, from time-domain pulse g t and frequency-domain pulse g f to modulation-symbol-domain pulse g x . time-domain receipt shaping pulse. s T ( t ) denotes the transmit baseband signal, gi ven by s T ( t ) = M − 1 X ℓ ′ = − L s [ ℓ ′ ] g t, T ( t − ∆ t ℓ ′ ) (2) where ∆ t = 1 / ( M ∆ f ) denotes the baseband sampling time interval. ∆ f and M denote the subcarrier interv al and o verall subcarrier number, respectiv ely . s [ ℓ ′ ] denotes the transmit baseband digital signal with length M + L , where L denotes the prefix length of the w av eform. g t, T ( t ) is the time-domain transmit shaping pulse. Introduce the following common as- sumptions based on Eq. (1): 1) The delay variation is considered as a linear model within the duration of baseband signal, that is, τ p ( t ) = τ p + γ p t (3) 2) The scaling effect of g t, T ( t ) caused by time-varying path delay τ p ( t ) can be ignored within the duration of baseband signal, that is, g t, T ( t − ∆ t ℓ ′ − τ p ( t )) ≈ g t, T ( t − ∆ t ℓ ′ − τ p ) (4) 3) The shaping mismatch caused by e − j 2 πf c τ p ( t ) between the transmit and receipt shaping pulses can be ignored within the duration of baseband signal, that is, g t, R ( t ) ∗  g t, T ( t − ∆ t ℓ ′ − τ p ) e − j 2 πf c τ p ( t )  ≈ ( g t, R ( t ) ∗ g t, T ( t − ∆ t ℓ ′ − τ p )) e − j 2 πf c τ p ( t ) = g t ( t − ∆ t ℓ ′ − τ p ) e − j 2 πf c τ p ( t ) (5) where g t ( t ) = g t, T ( t ) ∗ g t, R ( t ) denotes the overall time- domain shaping pulse. Thus, s R ( t ) can be further express as s R ( t ) = P − 1 X p =0 α p e − j 2 πf c τ p | {z } h p × M − 1 X ℓ ′ = − L s [ ℓ ′ ] g t ( t − ∆ t ℓ ′ − τ p ) e j 2 πν p t ! (6) where ν p = − f c γ p denotes the doppler frequency of the p - th path. The third assumption estabilishes for ν p << M ∆ f . Sampling the recei ved signal s R ( t ) at the time interv al of ∆ t while remo ving the prefix, the receipt baseband digital signal r [ ℓ ] is obtained, gi ven by r [ ℓ ] = P − 1 X p =0 h p e j 2 π M k p ℓ M − 1 X ℓ ′ = − L s [ ℓ ′ ] g t (∆ t ( ℓ − ℓ ′ − ℓ p )) ! = P − 1 X p =0 ℓ + L X ℓ ′ = ℓ − M +1 h p g t (∆ t ( ℓ ′ − ℓ p )) s [ ℓ − ℓ ′ ] e j 2 π M k p ℓ (7) where 0 ≤ ℓ < M , ℓ p = τ p / ∆ t ∈ R , k p = ν p / ∆ f ∈ R . Due to the tail attenuation property of g t ( t ) , there is g t ( t ) = 0 , | t | > ( G ∆ t ) / 2 (8) Thus, the receipt baseband digital signal r [ ℓ ] is further ex- pressed as r [ ℓ ] = P − 1 X p =0 ⌈ ℓ p ⌉ + G/ 2 X ℓ ′ = ⌊ ℓ p ⌋− G/ 2 h p g t (∆ t ( ℓ ′ − ℓ p )) s [ ℓ − ℓ ′ ] e − j 2 π M k p ℓ (9) where 0 ≤ ℓ < M . B. Input-output relationship in AFDM The transmit baseband digital signal s [ ℓ ′ ] of AFDM wave- form is gi ven by s [ ℓ ′ ] = 1 √ M M − 1 X m ′ =0 x [ m ′ ] e j 2 π  c 1 ℓ ′ 2 + m ′ ℓ ′ M + c 2 m ′ 2  , − L ≤ ℓ ′ < M (11) where x [ m ′ ] denots the m ′ -th transmit modulation symbol. It is seen that AFDM is a sort of chirp-based modulation wa veform. At receiv er side, the AFDM receipt modulation symbols y [ m ] are giv en by y [ m ] = 1 √ M M − 1 X ℓ =0 r [ ℓ ] e − j 2 π ( c 1 ℓ 2 + mℓ M + c 2 m 2 ) , 0 ≤ m < M (12) Substituting Eqs. (9) and (11) into Eq. (12), the modulation- symbol-domain input-output relationship of AFDM is ob- tained, giv en by Eq. (10), where g f ( m ) giv en by Eq. (13) is a sinc-like function, and [ H ] m,m ′ is the entry of the m -th row and m ′ -th column in channel matrix. g f ( m ) = sin ( π m ) M sin  π M m  e − j π ( M − 1) M m (13) From Eq. (10), it can be seen that the modulation symbols x [ m ′ ] have a con volution-lik e operation with the channel response. This implies that a single modulation symbol will 3 y [ m ] = 1 √ M M − 1 X ℓ =0 P − 1 X p =0 ⌈ ℓ p ⌉ + G/ 2 X ℓ ′ = ⌊ ℓ p ⌋− G/ 2 h p g t  ∆ t  ℓ ′ − ℓ p  1 √ M M − 1 X m ′ =0 x h m ′ i e j 2 π   c 1  ℓ − ℓ ′  2 + m ′  ℓ − ℓ ′  M + c 2 m ′ 2   e − j 2 π  c 1 ℓ 2 + mℓ M + c 2 m 2  = M − 1 X m ′ =0 x h m ′ i              e j 2 πc 2  m ′ 2 − m 2  P − 1 X p =0 ⌈ ℓ p ⌉ + G/ 2 X ℓ ′ = ⌊ ℓ p ⌋− G/ 2 h p e j 2 π M  M c 1 ℓ ′ 2 − m ′ ℓ ′  g t  ∆ t  ℓ ′ − ℓ p  g f  m −  m ′ −  2 M c 1 ℓ ′ − k p  | {z } g x,m ′ ,p ( m )              | {z } [ H ] m,m ′ (10) disperse into multiple weighted replicas after passing the channel, where the positions of the dispersed replicas are related to both path delays and Dopplers. Therefore, AFDM has the potential to obtain sufficient time-frequency div ersity gain when the delay and Doppler parameters of different paths leads the channel responses of these paths sufficiently separated in the modulation symbol domain. C. Pulse Shaping Mechanism in AFDM In Eq. (10), g x,m ′ ,p ( m ) denotes the p -th path response from the m ′ -th transmit modulation symbol at receiv er side. The magnitude env elop of g x,m ′ ,p ( m ) is decided by the modulation-symbol-domain shaping pulse. The expression of g x,m ′ ,p ( m ) in Eq. (10) shows that the modulation-symbol- domain shaping pulse is generated by a con volution-like operation betwenn the time-domain shaping pulse g t ( t ) and frequency-domain shaping pulse g f ( m ) . Fig. 1 presents the process to obtain the modulation-symbol-domain pulse g x,m ′ ,p ( m ) from the time-domain pulse g t ( t ) and frequency- domain pulse g f ( m ) . Firstly , the delayed time-domain pulse g t (∆ t ( ℓ ′ − ℓ p )) does a scale transformation with the factor of − 2 M c 1 and a followed circular shift of k p + m ′ . Next, each of the sampled time-domain pulse taps modulates with the frequency-domain pulse g f ( m ) . Finally , the modulation- symbol-domain shaping pulse g x,m ′ ,p ( m ) is formed after superposition and resampling on the m dimension. From the generation of g x,m ′ ,p ( m ) , it is inferred that the pulse tail in the modulation symbol domain is jointly suppressed by the time and frequenc y domain shaping pulses. I I I . P R O P O S E D P U L S E S H A P I N G M E T H O D F O R A F D M This section provides a modulation-symbol-domain pulse shaping method and the corresponding input-output relation- ship for AFDM wav eform. A. Pulse shaping method Based on the analysis in Sec. II-C, this section presents an AFDM modulation symbol pulse shaping method illustrated in Fig. 2 1 . Compared with the approach in [14], the proposed method allo ws more flexible pulse-width configuration of the time and frequency domain shaping pulses (e.g., whether to satisfy the Nyquist criterion in the frequency domain). 1 In practical zero-intermediate-frequency systems, time-domain pulse shap- ing is usually performed in the upsampled digital domain. For clarity of presentation, Fig. 2 depicts the time-domain pulse shaping in the analog domain. 2             , 0   <   -Point IFFT Cyclic -Prefix with Length  Cyclic -Postfix with Length        ,    <  +  Time - Domain Pulse Shaping, convolution with   ,               Cyclic -Prefix    Cyclic -Postfix  Ti m e - D om a i n P ul s e S ha pi ng, c onvol ut i on w i t h   ,   S a m pl i ng               , 0   <  +  C y c lic -P re fi x Re m ova l Ti m e -D om a i n O ve rl a ppi ng          , 0   <     -P oi nt FFT         +    D el ay  󰆒    C y c lic - P re fi x Re m ova l w i t h L e ngt h  Re m ove d S a m pl e s     Ti m e - D om a i n O ve rl a ppi ng          -P oi nt FFT (a) 2              , 0   <   -P oi nt I FFT C y c lic -P re fi x w i t h L e ngt h  C y c lic -P os t fi x w i t h L e ngt h         ,     <  +  Ti m e - D om a i n P ul s e S ha pi ng, c onvol ut i on w i t h   ,               C y c lic -P re fi x    C y c lic -P os t fi x  Time - Domain Pulse Shaping, convolution with   ,   Sampling             , 0   <  +  Prefix Removal Time -Domain Aliasing        , 0   <     -Point FFT         +    Delay  󰆒    Prefix Removal with Length  Removed Samples            Time - Domain Aliasing          -Point FFT (b) Fig. 2. Proposed modulation symbol pulse shaping method for AFDM at (a) transmitter side, (b) receiver side. T ime-domain pulse shaping is implemented through matched filtering at the transmitter and receiv er as the general real- ization in practical systems, while frequency-domain pulse shaping is achieved by time-domain windowing and aliasing at receiv er side. The combination of these two procedures realizes the pulse shaping in the modulated symbol domain for AFDM wav eform. The interpolation of a cyclic postfix together with the aliasing operation provides an adjustment margin for pulse-width in the frequency domain. Note that if the Nyquist shaping criterion is desired to be satisfied, the presented shaping method incurs shaping ov erhead in the time- frequency domain. When raised-cosine pulses are adopted in both domains, the overhead coefficients correspond to the roll-off factors in the respective domains. B. Corresponding input-output relationship The forms of the AFDM transmit baseband signal s [ ℓ ′ ] and receipt baseband signal r [ ℓ ] corresponding to the shaping method shown in Fig. 2 are consistent with Eq. (11) and Eq. (9), respectiv ely , except that the range of the time indexes ℓ ′ and ℓ are different, that is, − L ≤ ℓ ′ < M + W and 0 ≤ ℓ < M + W in the proposed method. Moreov er , the 4 y [ m ] = 1 √ M M + W − 1 X ℓ =0 G f [ ℓ ] r [ ℓ ] e − j 2 π  c 1 ℓ 2 + mℓ M + c 2 m 2  = e − j 2 πc 2 m 2 √ M        M − 1 X ℓ =0 G f [ ℓ ] r [ ℓ ] e − j 2 π  c 1 ℓ 2 + mℓ M  + W − 1 X ℓ ′ =0 G f h ℓ ′ + M i r h ℓ ′ + M i e − j 2 π   c 1  ℓ ′ + M  2 + m  ℓ ′ + M  M          = e − j 2 πc 2 m 2 √ M M − 1 X ℓ =0  G f [ ℓ ] r [ ℓ ] e − j 2 πc 1 ℓ 2 + G f [ ℓ + M ] r [ ℓ + M ] e − j 2 πc 1 ( ℓ + M ) 2  | {z } Time-Domain Aliasing , G f [ ℓ ]= r [ ℓ ]=0 for ℓ ≥ M + W e − j 2 π M mℓ (14)  ′  R ecei ved P i l ot P i l ot L eakage E ner gy  p , l eak P i l ot O ver al l E ner gy  p , al l G uar d I nt er val  = 2   1 ℓ  , m ax + 1 + 2  + − 1  R ecei ved D at a D2 P -I  = 2   1 ℓ  , m ax + 1 + 2  + − 1 = 2   , m ax +  + 1 ℓ  , m ax + 1 + 2  + − 1 G uar d f or D 2P -I G uar d f or P 2D -I G uar d f or P 2D -I G uar d f or D 2P -I  +  +  +  +   , ma x +    , ma x +    , ma x +    , ma x +  G uar d I nt er val  P2 D -I D2 P -I P2 D -I C H A N N E L  1 = 2   , m ax +  + 1 2  Fig. 3. Illustration of the embedded pilot for simulations (P2D-I: pilot-to-data interference, D2P-I: data-to-pilot interference). receipt modulation symbols y [ m ] are giv en by y [ m ] = 1 √ M M + W − 1 X ℓ =0 G f [ ℓ ] r [ ℓ ] e − j 2 π ( c 1 ℓ 2 + mℓ M + c 2 m 2 ) , 0 ≤ m < M (15) From s [ ℓ ′ ] , r [ ℓ ] and Eq. (15), the input-output relationship of the proposed shaping method is e xactly the same as the final form of Eq. (10), with the dif ference of g f ( m − ( m ′ − (2 M c 1 ℓ ′ − k p ))) = 1 M M + W − 1 X ℓ =0 G f [ ℓ ] e − j 2 π M ( m − ( m ′ − ( 2 M c 1 ℓ ′ − k p ))) ℓ (16) Additionally , Eq. (14) presents the relationship between y [ m ] and time-domain aliasing operation. I V . N U M E R I C A L R E S U LT S This section presents the numerical validations of the pro- posed shaping method in terms of the improvements of chan- nel sparsity and pilot-to-data interference using an embedded pilot located at m ′ = 0 as shown in Fig. 3. The time-domain Nyquist shaping pulse is decomposed into two root-raised- cosine pulses at both the transmitter and receiver with the roll-off factor of α t . The subcarrier number and length of cyclic-prefix are set to M = 4096 and L = 288 , respecti vely , common in 5G configurations. The chirp-rate parameters c 1 and c 2 are set to 2( ⌈ k p, max ⌉ + ζ )+1 2 M and 0 , respecti vely [5]. ζ is a parameter used to adjust the separation degree between channel paths considering the off-grid Doppler effect. The extended guard interval Ω + is adjustable for the tolerance for the interference caused by off-grid delay and Doppler . In the simulations, time-domain pulse shaping is implemented in the digital domain after upsampling, where the upsampling factor and filter order are set to 8 and 1024 , respectiv ely . In particular , when a Nyquist frequency-domain pulse shaping 4 (a) 4 (b) Fig. 4. Cumulativ e distribution functions of the highlight tap numbers P HL (the lower is better) for (a) using different shaping pulses with η = − 30 dB and ζ = 1 , (b) using rectangular window shaping with different ζ . with the roll-of f factor of α f is adopted, the length of cyclic- postfix W is set to ⌈ α f M ⌉ . There are P = 10 paths in the sim- ulated channels, where the path gains h p follow the complex Gaussian distribution of C N (0 , 1) , the path delays ℓ p follow the uniform distribution of U [0 , ℓ p, max ] and ℓ p, max = 10 , and the path Dopplers k p = k p, max cos θ p and k p, max = 1 , θ p ∼ U [ − π , π ] . The number of Monte Carlo tests is set to 10 4 . In the following figures, “Rect. ”, “Nyq.RC. ”, “Hamm. ” and “Cheb .70dB” denote the rectangular, Nyquist-raised-cosine, Hamming and Chebyshev (-70dB sidelobe) pulse shaping G f [ ℓ ] , respecti vely , in the frequenc y domain. A. Impro vement of the channel sparsity This subsection uses the number of “highlighted tap number P HL ” in the receipt modulation symbols y m as the criterion for ev aluating the channel sparsity improvement under different shaping pulses. The metric P HL corresponds to the number of power -dominated elements of y [ m ] beyond which the residual power becomes less than η times the total receiv ed power of y [ m ] . The parameters ζ and Ω + are set to 1 and 20 , respectiv ely . The cumulati ve distribution function (CDF) of the simulated P HL in Fig. 4-(a) shows that the use of Nyquist RC time-frequency shaping with higher roll-of f factors can significantly enhance channel sparsity , while the cost is the increasing of the time-frequency overhead. Howe ver , Fig. 4- (a) sho ws two counterintuitive issues: 1) When frequency-domain rectangular shaping is used, the increasing of time-domain roll-factor α t results in the slightly degradation of channel sparsity . This is caused by the inco- herent superposition of the overlapped path responses resultant from fractional Doppler . A smaller α t reduces the power dif- ference between adjacent samples of g t (∆ t ( ℓ ′ − ℓ p )) . These taps are further overlapped after being modulated with the high-sidelobe frequency-domain shaping pulse g f ( m ) in the 5 (a) (b) Fig. 5. Magnitude profiles of one realization of y [ m ] using different shaping pulses wih the subcarrier number of (a) M = 4096 , (b) M = 512 . modulation symbol domain. After the incoherent superposition in the modulation domain, the po wer variation of final path response may become more dramatical, resulting in some low- power elements in y [ m ] . This phenomenon results in slightly better channel sparsity in smaller α t . On the other hand, if the distance between the samples of g t (∆ t ( ℓ ′ − ℓ p )) is increased to alleviate the incoherent superposition effect in the modulation symbol domain, a higher α t can be beneficial for channel sparsity . Fig. 4-(b) provides the corresponding demon- stration, where the distance of the samples of g t (∆ t ( ℓ ′ − ℓ p )) increases in the modulation symbol domain via increasing ζ . And also, the incoherent superposition effect can be alleviated via decreasing the sidelobes of g f ( m ) which is already demonstrated in Fig. 4-(a) while the sparsity degradation caused by separated sidelobes. 2) The frequency-domain Chebyshev shaping results in the degradation of channel sparsity compared with the Hamming shaping which has higher proximal sidelobes. This is caused by the constant-sidelobe feature of the Chebyshev shaping. As sho wn in Fig. 5-(a), the far sidelobes using the Hamming shaping continue to attenuate, and become lower than those using the Chebyshe v shaping when a large subcarrier number is adopted. Meanwhile, the main energy of the path response using the Hamming shaping is more concentrated. This allo ws to achieve the same energy ratio with fewer elements of y [ m ] using Hamming shaping. B. Impro vement of the pilot-to-data interference This section uses av erage energy leakage ratio E p,leak E p,all to ev aluate the impact of different shaping pulses on pilot-to- data interference, where the range of E p,leak and E p,all can be seen in Fig. 3. The parameter ζ is set to 1. From Fig. 6- (a), it is seen that increasing α t can significantly reduce the pilot guard o verhead Ω + . Meanwhile, the use of Nyquist RC shaping in the frequency domain can significantly reduce the energy leakage floor , thereby achieving lower pilot-to-data interference under a giv en Ω + . This is due to the continuous and rapid attenuation features of the far sidelobes in the Nyquist RC shaping as shown in Fig. 5-(a). Additionally , in Fig. 6-(a), the performance curv es with the frequency-domain roll-off factors of 0 . 25 and 0 . 5 almost overlap. Similarly , Chebyshev shaping with lower near sidelobes performs worse than Hamming shaping in terms of ener gy leakage floor . The reason is that the Chebyshev shaping have higher far sidelobes 4 (a) 4 (b) Fig. 6. A verage leakage energy ratio E p,leak E p,all under different extended guard intervals Ω + with the subcarrier number of (a) M = 4096 , (b) M = 512 . than Hamming shaping when the subcarrier number is large. As a comparison, Fig. 6-(b) provides the ev aluation results for the case with the subcarrier number of 512 , indicating that Chebyshev shaping has a lower leakage level than Hamming shaping at this time. From Fig. 5-(b), it is seen that the far sidelobes of Chebyshe v shaping are always lower than Hamming shaping under a smaller subcarrier number . V . C O N C L U S I O N S This letter studies the input-output relationship of AFDM wa veform, sho wing the coupling between time-domain, frequency-domain, and modulation-symbol-domain shaping pulses. It points out that modulation-symbol-domain pulse shaping in AFDM can be achie ved by separately shaping in the time domain and frequency domain. Based on this analysis conclusion, a modulation-symbol-domain pulse shap- ing method which can achiev e the Nyquist pulse shaping is proposed for AFDM wav eform. Some numerical results validate the coupling relationship of pulse shaping and the effecti v eness of the proposed method in improving the channel sparsity and pilot-to-data interference. R E F E R E N C E S [1] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F . Molisch, and R. 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