A digital implementation of the PSK transmitter and receiver as multirate algorithms of the digital

1

A digital implementation of the PSK transmitter and receiver as multirate algorithms of the digital

signal processing

Krzysztof Świder, and Michał Papaj

Abstract—In this paper designs of the transmitter and the receiver of PSK signals as three-rate algorithms of the digital signal processing are described. Additionally we show the examples of the transmitter and the receiver of the following signals: BPSK, QPSK. We discussed also an influence of the cascade of the shaping filter and the receiving filter on the intersymbol interference in the received complex envelope.

Index Terms—PSK, BPSK, QPSK, π/4-QPSK, matched filter, Nyquist filter.

I. Introduction

The phase modulation, PSK, is one of the most known and commonly used digital modulation in modern tele- communication systems. The simplest variant of this mo- dulation is two-valued, binary phase modulation (BPSK - Binary Phase Shift Keying), which is, among others, used in systems with a direct-sequence spread spectrum. In sys- tems, where the higher transmissions rate is a necessity the major solution is a four-valued phase modulation QPSK (Quadrature Phase Shift Keying), π/4-QPSK or OQPSK (Offset Quadrature Phase Shift Keying). Signals with the four-valued phase modulation are commonly used in North American Digital Cellular System NADC, Personal Handyphone System PHS, Pacific Digital Cellular System PDC [1], [2], a satellite TV Digital Video Broadcast - Satellite (DVB-S) and many other telecommunication systems. In the literature there are many publications describing the transmitter and receiver of PSK signals in the analog time domain, however it lacks their descriptions in the discrete-time domain. Therefore a purpose of this article is to present transmitters and receivers of PSK signals as multirate DSP algorithms using for the descrip- tion operational calculus which have not been done that way yet. The paper is organized as follows. The section II describes the conceptional scheme of the PSK signals transmitter. Next, in the section III the implementation of the PSK receiver is discussed. The next sections, IV-A and IV-B discusses examples of transmitters and receivers of BPSK and QPSK signals.

K. Świder is with the Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 11/12 Gabrie- la Narutowicza Street, 80-233 Gdańsk-Wrzeszcz, Poland, e-mail:

krzysztof.swider@eti.pg.gda.pl

M. Papaj is with the Faculty of Health Sciences with Subfaculty of Nursing, 15 Tuwima Street, 80-210 Gdańsk, Poland, e-mail:

m.papaj@gumed.edu.pl.

II. The PSK transmitter

The block scheme of the digital triple-rate [3] PSK signal transmitter used in modern modems [4] is presented in Fig 1. In the transmitter we can observe three various sampling rates: the first one – bitrate called also transmission rate [bit/s], the second – baudrate called modulation rate [bod] and the third one – output rate [Sa/s], with which the quadrature modulation (QM) and digital to analog conversion are performed. Therefore, the presented transmitter is in fact a triple-rate DSP algorithm [3], [5].

d[i] encoder PSK c[k] c_L[n] Quadrature Modulatorw_S[n] u[n] x[n]

Quadrature Modulator with Interpolator

[bit/s] [symbol/s] [Sa/s]

L ↑ H

exp(jω_cn)

Fig. 1. A block scheme of the PSK transmitter

In Fig. 1 marked samples corresponds to following series:

real, marked with single-line arrows and complex - marked with doubled-line arrows. Respectively, starting from the input, symbols represent: d[i] – a bits series (transmitted data), c[k] – transmitted complex symbols series, L↑ – a zeroinserter (a time expander), c_L[n] – output of the time expander, HS – 1/L-bandpass baseband shaping filter (sender filter) [6], w_S[n] – a transmitted complex envelope [7], u[n] – a transmitted analytic signal [10] and x[n] – a transmitted real signal given on the channel input.

According to the presented scheme, Fig. 1 the first operation performed on a transmitted data is their trans- formation (encoding) into complex symbols. On this stage for each of log₂M -bit data series a complex symbol (a constellation point) belonging to the set of the M-point PSK constellation is assigned; c_m= exp(j(2πm/M + ϕ)) where m = 0, 1, ..., M − 1, ϕ ∈ [−π, π) and M = 2^l for l = 1, 2, 3, . . ..

The next operation is L-times sampling rate increment achieved by inserting L− 1 zero-valued samples between each two samples of the c[k] series

c_L[n] :=

 c[n/L]; n = 0,±L, ±2L, . . .

0 for the rest n for n∈ I, L ∈ N (1) (where I denotes the set of real numbers, N denotes the set of the natural numbers) and filtration of a such

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„condensed” series with the ideal (non-casual and with infinitely short transition band) 1/L-bandpass shaping filterH_S with a frequency response given by a formula:

H_S(e^jω) =

 = 0 for ω∈ (−π/L, π/L)

0 for ω /∈ (−π/L, π/L) (2) where ω ∈ [−π, π) is a normalized frequency given in radians per sample [rad/Sa]. (Because H_S(e^jω) is periodi- cal, with period equal to 2π in the normalized frequency domain ω, it is sufficient to analyze only the main period, which is range ω ∈ [−π, π) [8], [9]). In a result of a L- times interpolation of the symbol series we obtain a series of samples of the transmitted complex envelope {w_S[n]}.

The sampling rate of the complex envelope{w_S[n]} is L- times higher than the modulation sampling rate:

{wS[n]} = HS{L↑ {c[k]}} ; k, n ∈ I (3) where I is set of integer numbers, L↑ denotes abovemen- tioned expansion of a series of the length L.

The autocorrelation function of the series of statisti- cally independent and uniformly randomized transmitted symbols c[k] is given by the following formula [7]:

R_c[k₁, k₂] :=E {c[k₁]c^∗[k₂]} =

=

 E {|c[k1]|²} + (E {c[k1]})² for k₁= k₂ (E {c[k1]})² for k₁= k2 (4) where E { } denotes the operator of the expected value.

Additionally, assuming that {c[k]} is a random series of a zero-valued mean, E {c[k]} = 0, and a singleton variance (average power), E {|c[k]|²} = 1, we obtain the following formula for the autocorrelation function of the series{c[k]}:

R_c[k₁, k₂] = δ[k₁− k₂] (5) where δ[ ] denotes the Kronecker delta. Assuming, that Δ_k= k₁− k₂autocorrelation function formula (5) can be {c[k]} defined as follows:

R_c[Δ_k] = δ[Δ_k], where Δ_k:= k₁− k2; (6) Power spectral density (PSD) of the series {c[k]} (at least wide-sense stationary) is defined by:

P_c(e^jω) :=F∞{Rc[Δ_k]}; ω^∈ [−π, π) (7) whereF∞ denotes DTFT.

Considering the formula (6) for the autocorrelation function of the series{c[k]} we can observe that the PSD of the series is constant

P_c(e^jω) = 1 (8)

Thus, in result of a time expansion (3), a fullband power spectral density P_c(e^jω) of the series{c[k]} is transformed into power spectral density of the interpolated series (the transmitted complex envelope) w_S[n])

P_wS(e^jω) =

 |HS(e^jω)|²P_c(e^jLω) for ω∈ (−π/L, π/L)

0 for ω /∈ (−π/L, π/L)

(9)

where a normalized frequency corresponding to a L- times higher sampling rate stays in a range of ω ∈ (−π/L, π/L) – 1/L-band spectrum.

In contrast to the full-band PSD P_c(e^jω), 1/L-band spectrum P_wS(e^jω) can be shifted on the frequency axis ω: (P_wS(e^j(ω−ωc))) in a range of ω_c∈ (−π/L, π(1 − 1/L)) (where ω_cis a carrier frequency) using complex heterodyne – a multiplicator with two complex inputs, one for the transmitted complex frequency w_S[n] and one for the complex sinusoid (complexoid) e^jωcn. The outcome of the heterodyne block is the complex („analytic”) signal [10]:

u[n] = w_S[n]e^jωcn⇔ Pu(e^jω) = P_wS(e^j(ω−ωc)) (10) whose spectrum corresponds to the spectrum P_wS(e^jω) of the transmitted complex envelope w_R[n] heterodyned to the center frequency ω_c. A quadrature modulator together with the precedent interpolator forms a QMI block (Quadrature Modulator with Interpolator) [9], [11]

and gives on the output a real signal x[n] transmitted afterwards into a channel:

x[n] = Re(u[n]) = Re(w_S[n]e^jωcn) (11) III. The PSK receiver

The processing of the signal at the receiver end (Fig.2) is a transposition of the discussed in the section II a transmitter’s side processing (Fig.1).

r[n] w_R[n]

H R

[Sa/s] [symbol/s] [bit/s]

ˆ

c[k] d[i]ˆ

decoder and slicer

L ↓

exp(−jωcn)

Quadrature Modulator

Quadrature Demodulator with Decimation

Fig. 2. A coherent PSK receiver

In Fig. 2 the following notation is used: r[n] – a real signal at the output of a channel,HR – a receiving filter, w_R[n] – a received complex envelope, L↓ – a compressor of a series of a L multiplicity, ˆc[k] – an estimate of transmitted complex symbols and ˆd[i] – estimate of a transmitted binary series.

Assuming that the channel introduce only an additive white Gaussian noise (AWGN) [12], [13], the signal (real) received at the channel output is given by:

r[n] = x[n] + [n]; n∈ I (12) where [n] is a realization of an AWGN. The purpose of the processing in the PSK receiver is to obtain a series d[i] which is the best estimate of a transmitted (binary)ˆ data series, which is directly bind to obtaining a received complex envelope w_R[n], that is the best estimate of the transmitted one w_S[n].

The first algorithm of signal processing on the receiver side is a quadrature demodulation of the received series r[n]. The quadrature demodulation consists of two opera- tions: heterodyning the signal downward, to the baseband, in such way that its center frequency gets equal to zero, and further a filtration with a real lowpass receiving filter HR of a following frequency response:

H_R(e^jω)= 0 dla ω ∈ (−π/L, π/L) (13) In the result we obtain a received complex envelope w_R[n]:

w_R[n] =H_R{r[n]e^−jωcn} n∈ I (14) In order to obtain a series of estimates of transmitted complex symbols ˆc[k] we collect samples of the received complex envelope w_R[n] in time instances in which the complex envelope is the closest to the adequate transmit- ted symbols (in recovered symbols instants)

{ˆc[k]} = L↓ {wR[n]} = L↓

HR{r[n]e^−jωcn}

n, k∈ I (15) Assuming that there is the full synchronization between the transmitter and the receiver, the obtained by L-times decimation of the received complex envelope series ˆc[k] is given by [14]:

ˆ

c[k] = c[k]g_1/L[0] +

l=k

c[k]g_1/L[(k− l)L] + 1/L[k] (16)

where

{g1/L[k]} := L↓ {g[n]} (17) and

g[n] =

l=∞

l=−∞

h_R[l]h_S[n− l] (18) denotes the convolution of the impulse response h_S[n] of the shaping filter in the transmitter with the impulse response h_R[n] of the receiving filter in the receiver. The first component in the formula (16), depending only on the transmitted symbols, is a product of transmitted complex symbols c[k] with g_1/L[0]. The third component:

_1/L[k] = L↓ {HR{[n]}} (19) is a noise from the quadrature demodulator output at the output of the L-times decimator, whereas the second component of the formula 

l=kc[l]g1/L[(k− l)L] corre- sponds to intersymbol interferences (ISI) in the receiver [15], [7]. We can observe from the formula (16) that besides the noise component _1/L[k] the decisive influence on the received complex symbols estimate ˆc[k] belongs to the cascade of the shapingHSand the receiving filterHR. The cascade mentioned above will be denoted asG := HRHS

further in the paper.

In Fig.3 the simplified scheme (assuming ideal syn- chronization: carrier and symbol and lack of the noise in the channel) of interpolator (in the transmitter) and the decimator (in the receiver) is presented. We can

c[k] c_L[n] w_S[n] w_R[n] c[k]ˆ

L ↓ L ↑ H S H R

G

Fig. 3. A block scheme of the simplified QMI-QDD baseband cascade

observe, that according to the formula (16) in order to eliminate intersymbol interferences, which have the negative influence on the transmitted complex symbols estimate ˆc[k] (16), the cascadeG of the shaping filter H_S and the receiving one H_R has to fulfill the condition of (Nyquist condition [8]):

g_1/L[kL] =

 1 for k = 0

0 for k= 0 (20)

Having the condition fulfilled together with the lack of the noise in the channel, the estimate of transmitted complex symbols ˆc[k] is error-free.

ˆ

c[k] = c[k] (21)

The third component of the sum in the formula (16) forces an additional condition for the filters cascadeG , to ensure the best signal to noise ratio in the receiver in symbol instances [8]:

SNR = |_k=∞

k=−∞c[k]g[n− kL]|²_n=kL

E {|w[n]|²}_n=kL (22) The condition mentioned above is fulfilled when the receiving filter is matched to the shaping filter in the transmitter [16]:

H_R(e^jω) = H_S^∗(e^jω) (23) which implicates following relationship between the im- pulse response of the shaping filter h_S[n] and a receiving one (matched) h_R[n]:

h_R[n] = h^∗_S[−n] (24) In classical solutions both filters: sending HS and receiving HR have frequency response in shape of the square root raised cosine (SRRC) given by the formula [3]:

H^SRRC(e^jω) =

=

⎧⎪

⎨

⎪⎩

1, |ω| < ^(1−α)π_L

cos_π

4α

_Lω

π − (1 − α)

, ^(1−α)π_L  |ω|  ^(1+α)π_L

0, |ω| > ^(1+α)π_L

(25) Using in the cascade filters of the SRRC frequency response (Fig.4) effects with G being the Nyquist filter

of the frequency response of the raised cosine (RC) [17]:

G^RC(e^jω) =

=

⎧⎨

⎩

1, |ω| < ω₁

12

1 + cos_π

2α

_Lω

π − (1 − α)

, ω₁ |ω|  ω2

0, |ω| > ω2

(26) where ω₁ = ^(1−α)π_L , ω₂ = ^(1+α)π_L , whereas α∈ [0, 1] is called a bandwidth excess factor [18], [7] or roll-off factor [3].

0 0

a

ω

−2πL −π

L 2π

π L L L

2

√L

0 0

b

ω

−2πL −π

L 2π

π L L L2

L

Fig. 4. An amplitude characteristics (in the passband) of ideal filters: (a) (square raised root cosine) SRRC, (b) (Nyquist) RC, for following coefficientα values: (red α = 0, black α = 0.5, blue α = 1).

The last operation on the PSK receiver side is symbol detection and decoding of data basing on the series ˆ

c[k] (decoder decision statistics). The series ˆc[k] is put on the input of the multithreshold (complex) decision algorithm (slicer). At the output of the slicer the estimate of the transmitted symbols series is obtained. Further, the estimate is decoded and at the output we obtain a received data binary series ˆd[i].

There is many variants of the PSK modulation; to the classical ones [7], [19] belongs: Binary Phase Shift Keying (BPSK ) [20], Quadrature Phase Shift Keying (QPSK) [21], and π/4-QPSK [22]. All of the mentioned PSK signals can be obtained using a transmitter whose scheme is presented in Fig.1 and can be received using the receiver from the Fig.2.

IV. Examples of PSK modulators and demodulators A. BPSK modulator and demodulator

The simplest PSK modulation type is a two-state phase modulation (BPSK) in which the phase keying is done by generation of the complex envelope of M = 2 complex (real) symbols:

c_m∈ {exp(j0), exp(jπ))} = {1, −1} where m = 1, 2 (27) The abovementioned is realized by encoder in the following way:

c[k] = 2(d[k]− 0.5); k ∈ I (28) It is worth noticing that the modulation BPSK for each of symbols assigns log₂2 = 1 bit of data. That means, that the symbol rate (baudrate) is equal to the bitrate. In Fig. 7 the scheme of the transmitter is presented, whereas Fig.6 shows the block scheme of the receiver.

d[k] c[k] w_S[n] x[n]

encoder

−1/2



L ↑ H

exp(jω_cn)

QMI

[bit/s] [symbol/s] [Sa/s]

Fig. 5. A block scheme of the BPSK transmitter

-1 1

r[n] w_R[n]

H

L↓

exp(−jωcn)

1/2  +1/2

QDD

[bit/s]

[symbol/s]

[Sa/s]

Fig. 6. A block scheme of the BPSK receiver

The signal processing on the receiver side (Fig.6) consi- sts of (analogous to Fig.2) following stages: heterodyning the received signal r[n] to the baseband, L-times decima- tion (16), in result of which we obtain the estimate of the transmitted complex symbols:

{ˆc[k]} = {L↓ {H_R{r[n]e^−jωcn}}} (29) and decoding, which can be realized as follows:

d[k] = 0.5ˆˆ c[k] + 0.5 (30) In result we obtain at the output an estimate of the transmitted binary series.

−1.5 −1 0 1 1.5

−1.5

−1 0 1

1.5 a

real

imag

−20 −10 0 10 20

.5

−1 0 1

.5 b

n [Sa]

−1.5 −1 0 1 1.5

−1.5

−1 0 1

1.5 c

real

imag

−20 −10 0 10 20

.5

−1 0 1

.5 d

n [Sa]

Fig. 7. BPSK (shaping and receiving filter SRRC with coefficients:

L = 40, N = 6L + 1, α = 0.5): (a) transmitted complex envelope wS[n], (b) eye diagram Re(wS[n]), (c) received complex envelope wR[n], (d) eye diagram Re(wR[n]). Red dotes mark the series values in the symbol instant

In Fig.7 are presented examples of realizations of transmitted complex envelope w_S[n] and the received complex envelope w_R[n] on the Gauss plain together with

corresponding eye diagrams (of real components). Analysis of the presented figures leads to a conclusion that if there is in the transmitter a SRRC filter that introduces intersymbol interferences (ISI) into transmitted complex envelope w_S[n] (Fig.7.a, Fig.7.b), and the filer in the rece- iver is matched SRRC (24), then in the received complex envelope w_R[n] ISI does not occur(Fig.7.c, Fig.7.d), which proves the conclusions from the section (?ref).

B. QPSK modulator and demodulator

The four-state (quadrature) phase keying (QPSK) is a generation of a complex envelope of M = 4 complex symbols:

c_m∈ {exp(jπ/4), exp(j3π/4), exp(−j3π/4)), exp(−jπ/4)}

(31) where m = 1, . . . , 4. In this case for each of symbols log₂4 = 2 bits of data are assigned. In Fig.8 the block scheme of three-rate QPSK modulator is presented.

d[i] dI[k]

dQ[k]

cI[k]

cQ[k]

wI[n]

wQ[n] j

wS[n] x[n]

−1/2

−1/2 





L↑ L↑

√2

√2

HS

HS ^{Re( )}

z⁻¹ 2↓ 2↓

exp(jω_cn)

encoder

debiplekser QMI

[bit/s] [symbol/s] [Sa/s]

Fig. 8. A block scheme of the QPSK transmitter

In the QPSK transmitter transmitted data d[i] are split using a debiplexer (a serial-parallel converter) into two parallel path: in-phase (I) with the BPSK modulator and quadrature (Q) also with the BPSK modulator. In each of paths Cartesian components c_I[k] i c_Q[k] of the transmitted symbols series are interpolated by L-valued interpolation factor. In effect we obtain following series:

{w_I[n]} = H_S{L↑ {c_I[k]}} =

=√

2HS{L↑ {2↓ {d[i]} − 0.5}} i, k, n ∈ I (32) {wQ[n]} = HS{L↑ {cQ[k]}} =

=√ 2HS

L↑ {2↓ z⁻¹{d[i]} − 0.5}

i, k, n∈ I (33) where z⁻¹ is a one sample delay.

Further the output of the in-phase w_I[n] and the quadrature w_Q[n] component are put on the quadrature summation block. The resulting series is a transmitted complex envelope of the QPSK signal.

w_S[n] = w_I[n] + jw_Q[n] (34) It is worth noticing that binary series d_Q[k] and d_I[k] are processed separately in the in-phase and the quadrature path with a bitrate two times lower than the bitrate of the data series d[i] at the input of the QPSK transmitter.

Thus, the advantage of the QPSK modulation over the BPSK modulation is that having the same symbol rate we acquire two times faster transmission of the binary data

occurring at the input of the QPSK transmitter. The pair of bits at the output of the debiplexer is called a dibit.

The transmitted QPSK complex envelope, similarly as BPSK, is given at the input of the quadrature modulator consisting of: the complex multiplicator and the functor determining the real component of the analytic signal (10)). In result we obtain a real signal transmitted into a channel:

x[n] = Re(e^jωcnw_S[n]); n∈ I (35) In Fig.10.a and Fig.10.b examples of realizations of series occurring in the QPSK transmitter (Fig. 8) are presented. Similarly to the BPSK transmitter the shaping filter, if its impulse response is a SRRC (and here we assu- me that it is),HSin the transmitter purposely introduces an ISI (Fig.10). Intersymbol interferences can be observed in eye diagrams of particular Cartesian components of the transmitted complex envelope w_S[n] and also through an analysis of the trajectory w_S[n] – by occurrences of the characteristic dispersions in the neighborhood of nominal constellation points in symbol instances.

The QPSK receiver Fig.9 consist of (because of a simple interpretation of the QPSK modulation as a sum of two BPSK signals) BPSK demodulator, both in the in- phase and quadrature path. Thus the signal processing on the receiver side (Fig.9) consisting of heterodyning the signal to the baseband and L-times decimation (36, 37) is the inverse of the abovementioned processing on the transmitter side.

-1 1

-1 1

r[n]

HR

H_R ^{cˆI[k] dˆI[i]}

ˆ

c_Q[k] dˆ_Q[i]

L↓ L↓ Re( )

Im( ) exp(−jωcn)







√1 2

√1 2

1/2 1/2

d[i]ˆ 2↑

2↑ z⁻¹

biplekser

decoder and slicer QDD

[bit/s]

[symbol/s]

[Sa/s]

Fig. 9. A block scheme of the QPSK receiver

{ˆcI[k]} = L↓ HR

Re(r[n]e^−jωcn)

(36)

{ˆcQ[k]} = L↓ HR

Re(r[n]e^−jωcn)

(37) The recovered Cartesian components ˆc_I[k], ˆc_Q[k] of estimates ˆc[k] of transmitted complex symbols c[k] are given on the input of the detector and the symbol decoder.

On this stage series ˆc_I[k], ˆc_Q[k] are passed to the input of a biplexer (a parallel-serial converter) on the output of which we obtain estimates of the transmitted binary series:

dˆ_I[i]



=

 z⁻¹

 2↑

 1

√2ˆc_I[k] + 0.5



(38)

dˆ_Q[i]



=

 2↑

 1

√2ˆc_Q[k] + 0.5



(39)

Series ˆd_I[i] and ˆd_Q[i] at the output of the receiver are summed and in result the estimate of the transmitted data series is obtained:

d[i] = ˆˆ d_I[i] + ˆd_Q[i] (40) Similarly to the discussed previously BPSK receiver, in the QPSK receiver properties of the filters cascadeG has the same influence on the ISI occurrence.

a

real

imag

√1 2

√1 2

−1√ 2

−1√ 2 1.1

−1.1−1.1 1.1 0

0 −20 −10 0 10 20

b

n [Sa]

√1 2

−1√ 2 1.1

−1.1 0

c

real

imag

√1 2

√1 2

−1√ 2

−1√ 2 1.1

−1.1−1.1 1.1 0

0 −20 −10 0 10 20

d

n [Sa]

√1 2

−1√ 2 1.1

−1.1 0

Fig. 10. QPSK (the shaping and the receiving SRRC filer with coefficients:L = 40, N = 6L + 1, α = 0.5): (a) transmitted complex envelope wS[n], (b) eye diagram Re(wS[n]), (c) received complex envelopewR[n], (d) eye diagram Re(wR[n]). Red dots mark series values in symbol instants

In Fig.10.c and Fig.10.d are presented i.a. received envelope (complex if the cascadeG is a RC filter). From analysis of figures mentioned above we can conclude that when the pair of SRRC filters is used in the transmitter and the receiver, then the level of the ISI in the received complex envelope is negligibly small and comes from the finite length of filters impulse response. The ISI introduced in the transmitter (Fig.10.a) to the transmitted complex envelope w_S[n], if the shaping filterH_Sis the SRRC filter, are leveled in the received complex envelope w_R[n] by the receiving SRRC filterH_R.

V. Summary

In the paper the design of the transmitter (Fig.1) and the receiver (Fig.2) of PSK signals as three-rate DSP algorithms is discussed. Additionally, there are shown examples of the implementation of the transmitter and the receiver of signals: BPSK (sec.IV-A) and QPSK (sec.IV-B). Basing on the abovementioned designs the transmitter and the receiver for other modulations: the OQPSK (in which the quadrature component is delayed by the half of the symbol interval z^−L/2), the π/4-QPSK (by adding the difference encoder [19] on the transmitter side) and the M -PSK can be implemented.

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