LAB: Amplitude and Phase Shift Keying

8/12/2019 LAB: Amplitude and Phase Shift Keying

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ECEN 4652/5002 Communications Lab Spring 201404-07-14 P. Mathys

Lab 10: Phase and Hybrid Amplitude/Phase Shift Key-

ing, Carrier Sync

1 Introduction

A sinusoid like A cos(2f t+) is characterized by its amplitude A, its frequencyfand itsphase . Thus, after having seen amplitude shift keying (ASK) and frequency shift keying(FSK) as digital modulation methods, it is quite natural to also consider phase shift keying(PSK) to transmit digital data. An advantage of both FSK and PSK is that they areinsensitive to amplitude variations of the received signal. Because of its very nature, namelytransmitting information through phase changes, a PSK signal needs to be received with a

coherent receiver, as opposed to FSK and ASK that can also be received using non-coherenttechniques. The bandwidth requirements for a given bitrate, however, are smaller for PSKthan for ASK and FSK. In practice the use ofM-PSK is usually limited to M8 distinctphases. For larger M a combination of ASK and PSK, termed hybrid APSK in this labdescription, is generally used. One convenient way to implement hybrid APSK is to splitup the digital data into an in-phase and a quadrature data sequence, use pulse amplitudemodulation (PAM) for each and then feed the resulting continuous-time (CT) waveformsinto the in-phase and quadrature channels of a QAM (quadrature amplitude modulation)modulator.

1.1 Phase Shift Keying

Let an denote a DT sequence with baud rate FB = 1/TB. The following block diagram canbe used to first convert this sequence into a CT PAM signal s(t), which is then fed into aphase modulator (PM) to generate a PM signal x(t) at some carrier frequency fc.

PAMp(t)

PhaseModulator

an s(t) x(t)

Mathematically, the PM signal x(t) can be expressed as

x(t) =Ac cos

2fct+s(t)

, where s(t) = s(t) =

n=

anp(t nTB),

p(t) is a PAM pulse, and is a suitably chosen phase deviation constant. To demodulatea received PM signalr(t) produced by the above circuit, the blockdiagram shown below canbe used.

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2sin2fct

2cos2fct

LPF

at fL

LPFat fL

tan1wq(t)wi(t)

RcvrhR(t)

t= nTB

r(t) b(t)

vi(t)

vq(t)

wi(t)

wq(t)

s(t) bn

Note that, in order to cover the whole range from 0 to 2 (or to ) for (t), a four-quadrant version of tan1, such as atan2(wq,wi) in Matlab, has to be used. Next, assume

thatr(t) = cos

2fct+ s(t)

, with s(t) =

n=

anp(t nTB).

Thenvi(t) = 2 r(t) cos(2fct) = cos

s(t)

+ cos

4fct+ s(t)

,

vq(t) =2 r(t) sin(2fct) = sin

s(t)

sin

4fct+ s(t)

.

After lowpass filtering and sampling at time t= nTB this becomes

wi(t) = cos

s(t)

= wi(nTB) =wi[n] = cos

an

,

wq(t) = sin s(t) = wq(nTB) =wq[n] = sin an,where it is assumed that the PAM pulse p(t) satisfies Nyquists first criterion for no ISI, i.e.,

p(nTB) =

1, n= 0,0, otherwise.

= s(nTB) =

k=

akp(nTB kTB) =an.

The quantities wi[n] and wq[n] can be interpreted as the real and imaginary parts, respec-tively, of a point on the unit circle at angle an in the complex plane. If an can onlytake on Mdiscrete values, e.g., an {0, 1, . . . , M 1}, then the resulting M points canbe spread equally around the unit circle, each representing the transmission of a different

M-ary symbol. The following table shows the actual angle values used when

an {0, 1, . . . , M 1} , =2

M ,

and Mis a power of 2 in the range 21, . . . , 24.

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M an tan1

wq[n]/wi[n]

2 {0, 1} {0, }

4 /2 {0, 1, 2, 3} {0, /2, , 3/2}

8 /4 {0, 1, 2, . . . , 7} {0, /4, /2, 3/4, . . . , 7/4}

16 /8 {0, 1, 2, . . . , 15} {0, /8, /4, 3/8, . . . , 15/8}

Plottingwq[n] versuswi[n] for all Mvalues that an can take on results in graphs similar tothe ones shown below for M= 2 and M= 4.

wq[n]

wi[n]

wq[n]

wi[n]01

00

10

01

11

M=2 M=4

The graph which shows the locations of the M signal points and their geometrical rela-tionships with respect to each other is called a signal constellation. For M-ary phaseshift keying (PSK), the signal constellation consists ofMpoints spread equally around

a (unit) circle in a 2-dimensional signal space. Such constellations are also calledpolarsignal constellations (PSC). IfM is a power of 2, e.g., M = 2m, then each point in thesignal constellation corresponds to a particular pattern of m transmitted bits. There aremany ways in which bits can be associated with signal points, including Gray coding (sothat only one bit changes between adjacent signal points). For the purposes of these notesit is assumed that M-ary symbols are integers in the range 0, 1, . . . M 1 which correspondto their binary expansions, with the LSB written first. Binary representations for symbolsin an are shown for M= 2

1 . . . 24 in the table below.

M Values ofan Corresponding Binary Strings (LSB first)

2 {0, 1} {0, 1}4 {0, 1, 2, 3} {00, 10, 01, 11}

8 {0, 1, 2, . . . , 7} {000, 100, 010, 110, 001, 101, 011, 111}

16 {0, . . . , 6, 7, 8, 9, . . . , 15} {0000, . . . , 0110, 1110, 0001, 1001, . . . , 1111}

The signal constellations given previously only show the sampled in-phase and quadraturecomponentswi[n] andwq[n] at the outputs of the lowpass filters in the PM receiver. A more

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generalI-Q plot, which in addition also shows the transitions between the signal points, isobtained by also plotting wq(t) versuswi(t). The transition paths depend on how the PAMpulse p(t) is chosen. The following graph shows an I-Q plot for 8-PSK with a rectangularpulse p(t) of width TB = 1/FB where FB is the baud rate of the transmitted DT sequencean.

1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

IQ Plot for Mary PSK, M=8, ptype=rectx, rtype=pm, Eb/N

0= 60 dB

Inphase component wi(t)

Quadrature

componentwq

(t)

If p(t) is rectangular as in the above plot, then the transitions between signal points areessentially straight lines. If p(t) is triangular as shown in the next plot, then the phasechanges occur more gradual and the transistions between signal points all occur on the circleon which the signal points are located. Note that this implies that the amplitude of thetransmitted signal is constant (which is not the case for rectangular p(t)).

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1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

IQ Plot for Mary PSK, M=8, ptype=tri, rtype=pm, Eb/N

0= 60 dB


Quadraturecomponentwq

(t)

In a real communication system the received signal is usually noisy. In this case, the signalpoints (and the transitions between them) are scattered around their nominal locations. Anexample forM= 8, rectangular p(t), and a signal-to-noise ratio (SNR) ofEb/N0= 20 dB isshown in the following graph.

1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

IQ Plot for Mary PSK, M=8, ptype=rectx, rtype=pm, Eb/N

0= 20 dB


Q

uadraturecomponentwq

(t)

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Ifp(t) is triangular instead of rectangular, this changes as shown next.

1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

IQ Plot for Mary PSK, M=8, ptype=tri, rtype=pm, Eb/N

0= 20 dB


Quadraturecomponentwq

(t)

Assuming that the in-phase and quadrature noise components have equal power, and all Msignal points are equally likely, the decision rule at the receiver is to assume that the mostlikely transmitted signal is the one that is closest in (Euclidean) distance to the receivedsignal point. This leads to wedge-shapeddecision regions, similar to cutting up a birthday

cake into Mequal slices. A signal constellation for 8-PSK is shown in the left figure below.The corrspondingmaximum likelihood (ML)decision regions are shown in the graph onthe right, with the dashed lines representing the boundaries between decision regions.

wq[n]

wi[n]000

100

010

110

001

101

011

111

M=8

As the SNR decreases, some received signal points will eventually cross the decision regionboundaries, which in turn leads to symbol and bit errors at the receiver output.

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The next graph shows the PSD of a M-ary PSK signal withM= 8,fc = 4000 Hz, rectangularp(t), and a baud rate of FB = 250. The main signal energy is concentrated in the bandfc FB. . . f c+FB, but beyond that the signal energy decreases only relatively slowly, andleads to interference with other users in frequency division multiplexed systems.

0 1000 2000 3000 4000 5000 6000 7000 800060

50

40

30

20

10

0

f [Hz]

10log10

(S

x(f))[dB]

PSD, Px=0.49997, P

x(f

1,f

2) = 49.8141%, F

s=44100 Hz, N=44100, NN=2,

f=1 Hz

In practice, the tails of such a spectrum either have to be removed because of FCC (Fed-eral Communications Commission) regulations, or are removed by the transfer function ofthe transmission channel. In either case, just bandpass filtering the PSK signal leads tointersymbol interference (ISI) at the receiver, which can degrade the performance of thecommunication system, especially in the presence of noise. It is generally more desirable tocontrol the PSD of the PSK signal by choosing a suitable PAM pulse p(t), e.g., using a pulsewith raised cosine in frequency (RCf) spectrum, or root RCf (RRCf) spectrum. If this isdone by first converting an to a PAM signal s(t), which then modulates a PM transmitter,the result is a continuous phase modulation (CPM)signalx(t). The spectrum of sucha M-ary PSK signal with M = 8, fc = 4000 Hz, RCfp(t), and a baud rate ofFB = 250 isshown in the following figure.

0 1000 2000 3000 4000 5000 6000 7000 800060

50

40

30

20

10

0

f [Hz]

10log10

(S

x(f))[dB]

PSD, Px=0.50001, P

x(f

1,f

2) = 49.9997%, F

s=44100 Hz, N=44100, NN=2,

f=1 Hz

This can be demodulated using a phase detector, followed by a filter matched to p(t). Theblock diagram of such a receiver was shown previously. However, it turns out that this kindof receiver is suboptimum if p(t) is different from a rectangular pulse. To see why this is

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true, consider thePM-based PSK signal

x(t) = cos

2fct+ s(t)

= cos

s(t)

=wi(t)cos2fct sin

s(t)

=wq(t)

sin2fct ,

wheres(t) =

kakp(t kTB). Written out, the in-phase term is

wi(t) = cos

s(t)

= cos

k=

akp(t kTB)

=

k=

cos(ak)p(t kTB),

and the quadrature term is

wq(t) = sin

s(t)

= sin

k=

akp(t kTB)

=

k=

sin(ak)p(t kTB).

Because of the = statements above, PM-based PSK signals cannot be demodulated directlywith matched filters in the phase detector. Thus, the SNR is not maximized before the(nonlinear) tan1 function, which in turn leads to subpotimum performance in the presenceof noise.

Another approach that is very often used in practice is QAM-based PSKwhich splits theDT sequence an up into an in-phase component ai[n] and a quadrature component aq[n] bysetting

ai[n] = cos

an

, and aq[n] = sin

an

.

Each of these two signals is then converted separately into a PAM signal, using the samepulse p(t) in most cases. The two PAM signals are then fed into a quadrature amplitude

modulator (QAM) to form a PSK signal x(t) of the form

x(t) =

n=

ai[n]p(t nTB) cos 2fct

n=

aq[n]p(t nTB) sin2fct .

The blockdiagram of such a transmitter, with an {0, 1, . . . , M 1} and = 2/M, isshown below.

PAMp(t)

PAMp(t)

sin2an

M

cos2an

M

sin2fct

cos2fct

+

+

+ {0, 1, . . . ,M 1}an

ai[n]

aq[n]

si(t)

sq(t)

x(t)

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The advantage of using separate in-phase and quadrature components is that now a QAMreceiver like the one shown next can be used, where the LPFs are replaced by matchedfilters (MF), matched to p(t) (assuming an ideal transmission channel).

2sin2fct

2cos2fct

MF

hR(t)

MFhR(t)

tan1wq(t)wi(t)

M

2

t= nTB

r(t)

vi(t)

vq(t)

wi(t)

wq(t)

(t) bn

To make scatter plots and I-Q plots from this receiver, just simply plot wq(nTB) versuswi(nTB) or wq(t) versuswi(t), in the same way as this was done before with the PM receiver.

1.2 Hybrid Amplitude/Phase Shift Keying

Hybrid M-ary amplitude/phase shift keying (APSK) is a logical extension of QAM-basedM-ary PSK. For APSK both the amplitude and the phase of the carrier can be modulatedsimultaneously, resulting in QAM signal constellations. Using the blockdiagram below, aM-ary DT sequenceanis split up into aMi-ary in-phase sequenceai[n] and aMq-ary quadrature

sequenceaq[n], where Mi Mq = M. The two sequences are then converted to waveformsusing PAM with a pulse p(t) and combined into a QAM signal x(t) at carrier frequency fcusing a regular QAM modulator.

PAMp(t)

PAMp(t)

sin2fct

cos2fct

+

+

+

Mappingan

ai[n]

aq[n]

si(t)

sq(t)

x(t)

The signal constellation is obtained by plotting aq[n] versus ai[n]. The geometry of thesignal constellation is determined by the mapping from an to ai[n] and aq[n]. In principle,

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any 2-dimensional constellation, such as the two examples for M= 8 and M= 19 shownbelow, can be used as QAM signal constellation for hybrid APSK.

aq[n]

ai[n]

aq[n]

ai[n]

M= 8 M= 19

However, since in general a receiver as shown in the following blockdiagram is used, rectan-gular or cartesian signal constellations (CSC)are preferred because they lead to easierimplementations of the receiver.

2sin2fct

2cos2fct

MF

h(t)

MFh(t)

t= nTB

t= nTB

Quantizationand

InverseMapping

r(t)

wi(t)

wq(t)

vi(t)

vq(t)

wi[n]

wq[n]

bn

CSC constellations for M= 4 and M= 8 are shown in the next two figures. For M= 4 itis quite natural to use one bit per symbol for ai[n] and the other one for aq[n]. For M = 8it is less clear how three bits should be split up between the in-phase and quadrature datasequences. A somewhat arbitrary choice whenm in M = 2m is odd, is to use mi =m/2bits for the in-phase sequence and mq = m/2 bits for the quadrature sequence. The bit

assignments for each signal point were made as follows. The firstmi out of everym bits areassigned to the in-phase sequence ai[n], with the LSB coming first (in the leftmost position).The remaining mq out of every m bits are similarly assigned to the quadrature sequenceaq[n], again with the LSB coming first.

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00 10

01 11

aq[n]

ai[n]

000 100 010 110

001 101 011 111

aq[n]

ai[n]

M= 4 M= 8

Apart from the usual problem of synchronizing the receiver with the transmitter, the mainproblem of the receiver is to associate the received noisy sample pairs ( wi[n], wq[n]) withthe most likely transmitted signal point (ai[n], aq[n]). A maximum liklelihood (ML) receivermeasures the Euclidean distance between (wi[n], wq[n]) and all possible (ai[n], aq[n]), andoutputs that value ofbn that corresponds to the signal point which is closest to the received

point. The graph below shows some received signal points for a noisy QAM signal when aM= 4 CSC constellation is used.

2 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0

0.5

1

1.5

2

IQ Plot for Mary APSK, M=4, ptype=rect, Eb/N

0= 10 dB


Quadraturecompon

entwq

(t)

The horizontal and vertical dashed lines represent the decision boundaries for a ML receiver.Because they are orthogonal, decisions on the in-phase and quadrature components can bemade independently without loss of optimality, which simplifies the decision and quantiza-tion process significantly. The next two figures show the bit assignments and the decisionboundaries for a M= 16 QAM-CSC signal.

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aq[n]

ai[n]

0000 1000 0100 1100

0010 1010 0110 1110

0001 1001 0101 1101

0011 1011 0111 1111

M= 16

Telephone modems use similar constellations with up to about 13 bits per symbol. The lim-iting factor in this case is the quantization noise which is introduced by the A/D conversionof the analog telephone signal at the central office.

1.3 Carrier Synchronization

In addition to extracting symbol rate information from the received signal r(t), it is alsonecessary to extract carrier synchronization information at a PSK or APSK receiver. In adigital receiver the starting point is to convert the received real bandpass signal to a complex-valued lowpass signal, so that the sampling rate for further processing can be minimized.

Assume that the receiver uses cos(2fct) for the carrier reference, but the transmitter actuallyused cos(2fct+ e(t)), where e(t) is a time-varying phase error, as carrier. In complexnotation the received signal is of the form

r(t) = Re{sL(t) ej(2fct+e(t))}=

sL(t) ej(2fct+e(t)) +sL(t) e

j(2fct+e(t))

2 ,

wheresL(t) is a complex-valued baseband PAM signal. For CPM M-PSK with data dn

r(t) = cos

2fct+e(t) + s(t)

,

and thus

sL(t) = cos(s(t)) +j sin(s(t)) =ejs(t) , =

2

M

,

where

s(t) =

n=

dnp(t nTB), dn {0, 1, . . . , M 1} ,

andp(t) is a real-valued PAM pulse. For QAM with in-phase data di[n] and quadrature datadq[n]

r(t) =si(t) cos(2fct+e(t)) sq(t) sin(2fct+e(t)),

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and thereforesL(t) =si(t) +j sq(t),

with

si(t) =

n=di[n]p(t nTB), and sq(t) =

n=dq[n]p(t nTB).

For QAM-basedM-PSK with data sequencedn {0, 1, . . . , M 1}

di[n] = cos2dn

M

, and dq[n] = sin

2dnM

,

and thus

sL(t) =

n=

di[n] +j dq[n]

p(t nTB) =

n=

ej2dn/Mp(t nTB),

wherep(t) is a real-valued PAM pulse.The following block diagram can be used to convert the received real bandpass signal r(t)to a complex-valued lowpass signal rL(t).

2 ej2fct

LPFor MF

r(t) v(t) rL(t)

After multiplication by the complex exponential

v(t) = 2 r(t) ej2fct =

sL(t) ej(2fct+e(t)) +sL(t) e

j(2fct+e(t))

ej2fct

=sL(t) eje(t) +sL(t) e

j(4fct+e(t)) ,

and thus, after lowpass filtering,

rL(t) =sL(t) eje(t) .

Thus, for CPM-based M-PSK,

rL(t) = expj

n=

2 dnM

p(t nTB) eje(t) , dn {0, 1, . . . , M 1} ,

and, for QAM-basedM-PSK,

rL(t) =

n=

ej2dn/Mp(t nTB) eje(t) , dn {0, 1, . . . , M 1} .

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If there is no ISI and p(nTB) =n, then the samples at t = nTB are in both cases

rL(nTB) =ej2dn/Meje(nTB) .

Sincedntakes on integer values, the first (data-dependent) term can be eliminated by raising

rL(nTB) to the M-th power, i.e.,

rML (nTB) =ejMe(nTB) .

This leads to the following blockdiagram for estimating first e(nTB) and then (assuminge(t) is bandlimited to FB/2) e(t) by using PAM with a sinc pulse p(t).

(.)M arg(.)

M

PAMp(t)

t= nTB

rL(t) rL(nTB) rM

L(nTB) e(nTB) e(t)

Note that for this to work it is crucial that the sampling times t = nTB are estimatedaccurately. The see how this can be done for QAM-based signals, start from

rL(t) =

si(t) +j sq(t)

eje(t) ,

where both si(t) and sq(t) are real-valued PAM signals. Computing the magnitude squaredyields

|rL(t)|2 =s2i (t) +s

2q(t).

Except for PAM signals with rectangular p(t), this has a spectral component at FB which canbe filtered out and used to synchronize the sampling circuit at the receiver. For CPM-basedsignals, start from

rL(t) =ejs(t) eje(t) ,

wheres(t) is a real-valued PAM signal. In this case |rL(t)|2 will not have a useable spectralcomponent at FB. However, the first derivative with respect to t is

drL(t)

dt =j

d

dt

s(t) +e(t)

ejs(t) eje(t) .

Taking the magnitude squared yields

drL(t)dt

2 = ds(t)dt

+de(t)

dt

2,

which in most cases contains a spectral component at FB. Note that, if e(t) is a slowlyvarying phase error, then de(t)/dt 0.

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2 Lab Experiments

E1. M-ary PSK. (a)Write a Matlab function, calledpskxmtr, which generates anM-PSKsignal from an M-ary data sequence dn (taking on values in {0, 1, . . . , M 1}) with baudrateFB. Depending on the value of xtype, the function should either generate PM-based or

QAM-based PSK signals. The header for the pskxmtr function is given below.

function [x,t] = pskxmtr(M,dn,FB,Fs,ptype,pparms,xtype,fcparms)

%pskxmtr M-ary Phase Shift Keying (PSK) Transmitter for PM-based

% (pm) and QAM-based (qam) PSK signals

% >>>>> [x,t] = pskxmtr(M,dn,FB,Fs,ptype,pparms,xtype,fcparms)


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function [bn,wt,ixn] = pskrcvr(M,t,r,rtype,fcparms,FBparms,ptype,pparms)

%pskrcvr M-ary Phase Shift Keying (PSK) Receiver for PM-based

% (pm) and QAM-based (qam) Reception of PSK signals

% >>> [bn,wt,ixn] = pskrcvr(M,t,r,rtype,fcparms,FBparms,ptype,pparms)


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effect of the BPF on the PSK signal.

(d) Repeat (c), but this time let p(t) be an RCf pulse with = 0.5 and k 10. Whatconclusions can you draw from comparing the results in (c) and (d)?

E2. Hybrid ASK/PSK. (a) Here is the header for a Matlab function called apskxmtr

which is used to generate M-ary APSK CSC (amplitude/phase shift keying, cartesian signalconstellation) signals for M = 2m. If m is even, then the signal points are arranged ona square grid, otherwise a rectangular grid is used with the longer egde in the in-phasedirection. The nominal signal constellation points are at the intersections of1,3, etc, inthe in-phase and quadrature directions.

function [x,t] = apskxmtr(M,dn,FB,Fs,ptype,pparms,fcparms)

%apskxmtr M-ary (M=2^m) Hybrid Amplitude/Phase Shift Keying (APSK)

% Transmitter for Cartesian Signal Constellations (CSC)

% >>>>> [x,t] = apskxmtr(M,dn,FB,Fs,ptype,pparms,fcparms)


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It is assumed that the received signalris scaled properly so that the nominal signal points areat the intersections of1,3, etc, of the demodulated in-phase and quadrature components.

function [bn,wt,ixn] = apskrcvr(M,t,r,fcparms,FBparms,ptype,pparms)

%apskrcvr M-ary (M=2^m) Hybrid Amplitude/Phase Shift Keying (APSK)% Receiver for Cartesian Signal Constellations (CSC)

% >>>>> [bn,wt,ixn] = apskrcvr(M,t,r,fcparms,FBparms,ptype,pparms)


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function an = bin2m(dn,m)

%bin2m Binary (LSB first) to M-ary symbol conversion for M=2^m

% >>>>> an = bin2m(dn,m) > dn = m2bin(an,m)


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undoing the scaling, look at eye diagrams or scatter plots of the signals received from thewav files.

(b) Analyze and demodulate the signals in the files apsksig1001.wav, apsksig1002.wav,and apsksig1003.wav. All signals are (A)PSK transmissions which contain English ASCIItext and use either a rectangular p(t) of width TB or a RRCf p(t) with = 0.5. Whendemodulating these signals, remember that the time axis starts at t = TB/2, and makesure you choose the right scaling for the received signal so that it fits the decision boundariesof your quantizer and inverse mapping. Look at eye diagrams or scatter plots to determinethe correct scaling factors.

c20002014, P. Mathys. Last revised: 04-07-14, PM.

20

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LAB: Amplitude and Phase Shift Keying

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