Spectral Characteristics of Digitally Modulated Signals

Spectral Characteristics of Digitally Modulated Signals 16:332:546 Wireless Communication Technologies Spring 2005

Lecture7&8

Department of Electrical Engineering, Rutgers University, Piscataway, NJ 08904 Taught by Dr. Narayan Mandayam [email protected] Documented by Baozhen Yu [email protected]

± j ( 4πf c t + 2πf cτ + 2φT )

Abstract: This lecture first introduces the standard representation of complex baseband signal of digital modulated signal and its power spectral density (PSD). Then it shows how the PSD of bandpass signals can always be obtained by its complex baseband envolope. Finally, two important modulation schemes OFDM and PSK are introduced and the spectral characteristics of these two modulated signals are discussed.

Observe that E[ e

I.

As svv () is real and even, we got

PSD of Modulated Bandpass Signal

As we know that bandwidth efficiency is a very important factor for choosing a modulation scheme. To learn the bandwidth efficiency of one modulation scheme, we need to get the power spectral density of the corresponding modulated signal. Generally, a digitally modulated signal can be written as

s(t ) = Re{v (t )e j ( 2πfct +φT ) } ,where v(t) is the baseband equivalent complex envolope and φT is the random phase. To calculate the PSD of s(t), we can rewrite it as:

1 s(t ) = {v (t )e j ( 2πfct +φT ) + v * (t )e − j ( 2πfct +φT ) } 2 Then the PSD of s(t) can be derived by:

We got

1 4

1 E[v (t + τ )v(t )e j ( 4πfct +2πfcτ +2φT ) 4 + v * (t + τ )v(t )e − j 2πfcπ + v(t + τ )v * (t )e j 2πfcπ + v * (t + τ )v * (t )e j ( 4πfct +2πfcτ +2φT ) ]

1 4

φss (τ ) = φ *vv (τ )e − j 2πf τ + φvv (τ )e j 2πf τ As,

c

c

φvv (τ ) = φ *vv ( −τ ) , we got,

1 sss ( f ) = [ svv ( f − f c ) + svv * ( − f − f c )] 4

1 sss ( f ) = [ svv ( f − f c ) + svv ( f + f c )] 4 So, the PSD of bandpass signal s(t) is completely determined by the PSD of its complex baseband envelope v(t). As a consequence, the results of performance of various modulation and demodulation techniques are independent of carrier frequencies and channel frequency bands. II.

Standard Representation of Complex Baseband Signal

1. Standard Format: We now move on to study the standard representation of complex baseband signal of a digitally modulated signal. Generally, the baseband signal v(t) can be written as:

v (t ) = A∑ b(t − kT , xk )

φss (τ ) = E[ s(t + τ ) s(t )]

=

]=0

k

,where

A

is

the

carrier amplitude, xk = ( xk , xk −1 , xk −2 ,..., xk − K ) is the source

symbol sequence, K is the memory length, which depends on modulation scheme, T is the symbol duration, and b( t − kT , xk ) is an equivalent shaping function of duration T.

This is the standard representation of the complex baseband signal. 2. An Example of Amplitude-Shift Keying (ASK) Modulation For ASK modulation, we have

v ( t ) = A ∑ x n ha ( t − nT )

1 2

φvv (t + T + τ , t + T ) = E[v (t + T + τ )v * (t + T )] A2 ∑∑ E[b(t +T + τ − iT , xi )b * (t + T − kT , xk )] 2 i k Let i'=i-1 and k’=k-1 =

φvv (t + T + τ , t + T )]

n

A2 2

∑∑ E[b(t +τ − i' T , x

,where {xn } = {xn + jxn } is the source

=

symbol sequence, ha (t ) is the amplitude shaping pulse ( square wave or something else)

Assume the source symbols be stationary, i.e. xi = xi +1 , then, we have

I

Q

Here, k=0 which means no memory. And the equivalent shaping function:

i ' +1

) b * ( t − k ' T , x k ' +1 )

k'

φvv (t + T + τ , t + T ) A2 2

=

b(t , xk ) = xk ha (t )

i'

∑∑ E[b(t +τ − i' T , x i'

i'

)b * (t − k ' T , x k ' )]

k'

The bandpass signal s(t) can be written as:

= φvv (t + τ , t )

s(t ) =

So, v(t) is cyclostationary. Thus, φvv (τ ) can be obtained by taking the time average of φvv (t + τ , t ) i.e. φvv (τ ) =

A∑ {| xn | ha (t − nT ) cos[2πf c t + arg( xn )]} n

,where | x n |=

( x nI ) 2 + ( x nQ ) 2 −1

and arg( xn ) = tan ( III.

A2 1 T E[b(t +τ − iT , xi )b * (t − kT , xk )dt ] ∑∑ 2 i k T ∫0

Q n I n

x ) x

Change variable: t-kT=z, we got

PSD of Baseband Signal

As we have given the standard representation of the baseband complex envelope of the modulated bandpass signal. We can use it to get the PSD of the baseband signal .

∑ b(t − kT , x

Recall: v ( t ) = A

k

)

k

A2 2T

=

1 E[v (t + τ )v * (t )] 2

A2 ∑∑ E[b(t +τ − iT , xi )b * (t − kT , xk )] 2 i k

Claim: v(t) is a cyclostantionary process, i.e. φvv (t + τ , t ) is periodic in t with period T Proof:

i

− kT +T

− kT

k

A2 2T

∑∑ ∫

− kT +T

− kT

m+ k k

A2 2T

=

∑∫

−∞

−∞

m

E[b( z +τ − (i − k )T , xi )b * ( z, x k )dz

φvv (τ ) =

Let i-k=m,

Then its autocorrelation is given by:

φvv (t + τ , t ) =

∑∑ ∫

φvv (τ ) =

E[b( z +τ − mT , x m+ k )b * ( z, x k )dz

E[b( z + τ − mT , x m )b * ( z, x0 )dz

So, its PSD is given by

S vv ( f ) = E[

A2 2T

m

A2 = E[ 2T ⋅∫

−∞

−∞

−∞

−∞

−∞

−∞

∑∫ ∫

∑∫ m

−∞

−∞

b( z + τ − mT , xm )b * ( z, x0 )dz ⋅ e − j 2πfcτ dτ

b( z + τ − mT , xm ) ⋅ e − j 2πfc ( z +τ − mT ) dτ ]

[b * ( z, x0 ) ⋅ e j 2πfc z dz ⋅ e − j 2πfc mT ]

A2 2T

= ⋅∫

−∞

−∞

=

∑ m

E[ ∫

−∞

−∞

2

[ b * ( z , x0 ) ⋅ e 2

A 2T

si (t ) = E A xiI φ1 (t ) + E A xiQφ2 (t ) ,

b(τ ' , xm ) ⋅ e − j 2πfcτ ' dτ '

∑

j 2πf c z

dz ⋅ e

− j 2πf c mT

where E A = A T

]

2

is the symbol energy in

the waveform.

E[ B( f , xm ) B * ( f , x0 )]e − j 2πfc mT

m

,where B ( f , x m ) is the Fourier transform of

b(t , xm ) Observations: The PSD of v(t) depends on:

One popular ASK is M-QAM with source xi = xiI + jxiQ , where symbols as:

{±1,±3,±5,...,±( N − 1)} and

xiI , xiQ ∈ N= M.

For the case of 16-QAM, constellation is shown in Fig 1

a) The form of the equivalent pulse shaping fucntion b( t , xm ) .

the

signal

φ2 ( t )

b) The correlation properties of source sequence xm .

2 EA

φ1 ( t )

While the above gives a frequency domain representation or characteristics for a specific candidate v(t). What about the representation of whole class of feasible waveforms? Generally, v(t) during a symbol interval belong M to a {vl ( t )}l =1 , represented in terms a set of orthonormal basis functions N> δτ ( the RMS delay spread). So, the channel will look like a flat fading channel and the need for equalization is avoid. Also, as each source symbol in the block of length N is transmitted in parallel by employing N orthogonal subcarriers, the symbol rate on each subcarrier is much less than the serial source rate. As a result the effect of delay spread is reduced. These are the advantages of OFDM.

j 2π ( n −

n =0

T

= ha (t )∑ xk ,n exp{ ,

N −1 )t 2 }

xk = {xk ,0 , xk ,1 ,..., xk , N −1}

3. Why OFDM attractive? OFDM modulation is attractive because it can be achieved by using either inverse discrete Fourier transform (IDFT) or inverse fast Fourier transform (IFFT). Consider k=0 and ignore the frequency effect

j 2π ( n − N − 1 )t 2 } . Also choose T ha (t ) = uT (t ) , we have

term exp{

N −1

v (t ) = A∑ x0,n exp{ j n =0

2. Representation of OFDM signal

2πnt } , 0 ≤ t ≤ T = NTs T

If we sample v(t) at t=kTs, we got

The complex envelope of OFDM modulated signal is:

N −1

v ( kTs ) = A∑ x0,n exp{ j n=0

N −1

v (t ) = A∑∑ xk ,nφn (t − kT ) , k

N −1

N −1

= A∑ x0,n exp{ j

n =0

,where {φn ( t )} are orthogonal waveforms

N −1 )t 2 chosen as φn (t ) = ha (t ){ j } , T n=0,1,…,N-1. And ha ( t ) = uT ( t ) , which is a rectangle pulse. As the frequency separation is 2π ( n −

1 , {φn (t )} are orthogonal. T

n =0

2πnkTs } T

2πnk } N

Denote {x0,k } = {v ( kTs )} , k=0, 1, … , N-1 Then {x0,k } are just the IFFT of the block

A x0 , where x0 = ( x0,1 , x0, 2 ,..., x0,N −1 ) So, the transmitter is easy to implement. Fig 3 shows the scheme of an OFDM transmitter.

We can see that at time instant k, N source symbols are transmitted using the N distinct subcarriers.

xk ,n are

Usually,

chosen

from

QAM

constellation. The standard format is then given by

v (t ) = A∑ b(t − kT , xk ) , k

Fig 3 A OFDM transmitter 4. PSD of OFDM Signal Assume the source symbols are zero mean and the amplitude shaping pulse is ha (t ) , then the PSD of s(t) is given by

shaping pulse, M is the size of alphabet,

S vv ( f ) 2

A2 2 N −1 N −1 1 = δ x ∑ | H a ( f − (n − )) | T T 2 n =0 , where

δ x2 =

xk = {±1,±3,...,±( M − 1)} Fig 5 shows the signal constellation of the 8PSK,

1 E[| xk ,n |2 ] 2

If ha ( t ) = uT ( t ), then H a ( f ) = sin c( f ) . Fig 4 shows the PSD of s(t) in this case:

Fig 5 Constellation of 8PSK Often

we

choose

h p (t ) = uT (t )

and

ha (t ) = uT (t ) or raised cosine pulse. Fig 4 PSD of OFDM signal We can see from the figure that the main lope carries more part of the energy as N increases. This implies that we can get better spectral efficiency as N increases. V. 1

Phase Shifting Keying (PSK)

What’s PSK

2.

PSD of PSK signal

Let’s assume uncorrelated source symbols and h p (t ) = uT (t ), Also we assume source symbols are equal probable and defined by set:

xk ∈ {2i − 1 − M : i = 1,2,... M } Then b( t , xk ) = ha ( t ) exp{ j

π M

xk h p (t )} ,

PSK is perhaps the generic form of modulation most widely utilized in contemporary practice, ranging from voice-band modems to high-speed satellite transmission. As the name suggest, the signal set is generated by phase modulation of a sinusoidal carrier to one of M equispaced phase positions.

The result followed from the following:

For M-ary PSK signal, the standard format of its complex baseband signal is given by

E[exp{ j

v (t ) = A∑ b(t − kT , xk ) ,

b(t − kT , xk ) = ha (t ) exp{ j

∆

,where sin f =

π M

π

h p (t ))

M

sin( Mx ) M sin x

xkα (t )}] = sin f (

π M

α (t ))

Recall: S vv ( f ) =

k

where

And E[b( t , xk )] = ha ( t ) sin f (

π M

xk h p (t )} ,

where xk = xk (no memory), ha (t ) is the amplitude shaping pulse, h p (t ) is the phase

A2 2T

= ⋅∫

−∞

−∞

∑ m

E[ ∫

−∞

−∞

b(τ ' , xm ) ⋅ e − j 2πfcτ ' dτ '

[b * ( z, x0 ) ⋅ e j 2πfc z dz ⋅ e − j 2πfcmT ]

For uncorrelated source symbols, we have

S vv ( f ) = A2 2T A2 = 2T

=

A2 = 2T

−∞

−∞

−∞

−∞

−∞

−∞

−∞

−∞

−∞

−∞

−∞

−∞

∫ ∫

∫ ∫ ∫ ∫

E[b(τ ' , x0 ) ⋅ b * ( z, x0 ) ⋅ e − j 2πfc (τ ' − z ) dzdτ '

e

j

π M

[ h p (τ ')− h p ( z )] x0

t − >∞

M

M sin{

sin πt

As lim

π

sin{M

M sin(

π

M

ha (τ ' )h a ( z )dzdτ '

[h p (τ ' ) − h p ( z )]} [h p (τ ' ) − h p ( z )]}

ha (τ ' )h a ( z )dzdτ '

dt = 1

π

M

t)

Fig 6 PSD of PSK with different M

We have S vv ( f ) =

A2 = 2T =

−∞

−∞

−∞

−∞

∫ ∫

Table 1

ha (τ ' )ha ( z ) ⋅ e

− j 2πfc (τ ' − z )

dzdτ '

2

A | H a ( f ) |2 2T

Bandwidth efficiency of PSK

M

2

4

8

16

32

64

ηB

0.5

1

1.5

2.0

2.5

3.0

Choose h p ( t ) = uT ( t ), then

A2 sin(πfT ) 2 A2T sin(πfT ) 2 S vv ( f ) = [ ] = [ ] 2T 2 πf πfT sin(πfT ) 2 = E0 [ ] πf ,where

E0

T = (log

is the symbol energy and M 2

)T b .

For a fair comparison of bandwidth efficiency with different M, we substitute T with (log 2M )Tb and get:

S vv ( f ) =

A2 (log 2M )Tb sin(πf (log 2M )Tb ) 2 [ ] 2 πf (log 2M )Tb

Fig 6 shows the PSD of the PSK signals with different M. Here, the bandwidth efficiency is defined as

From the table, we can see that the bandwidth efficiency η B increases as M increases for M-ary PSK. However, its the power efficiency decreases as M increases due to the closer distant between different signals in the constellation.

ηB =

Rb and the bandwidth B B

here is defined as the “null-to-null” bandwidth. From the figure, we can get table 1 which shows the bandwidth efficiency of PSK with different M.

VI.

Conclusion

In this lecture, the PSD of modulated signals are discussed. First, we showed that the PSD of bandpass signal s(t) is completely determined by the PSD of its complex baseband envelope v(t). Then two important modulation schemes, OFDM and PSK, are introduced. OFDM modulation is not only good to fight multi-path fading but also easy to implement, making it very attractive for high bit-rate wireless applications in a multipath radio environment. M-ary PSK is also widely used in contemporary practice, ranging from voice-band modems to high-speed satellite transmission. Reference: [1] T. Rappaport, Wireless Communications.

Principles and Practice. 2nd Edition, Prentice-Hall, Englewood Cliffs, NJ: 1996. [2] J. Proakis, Digital Communications, 4th Edition, McGraw-Hill, NY: 2002. [3] S. G. Wilson, Digital Modulation and Coding. Prentice-Hall, 1996 [4] Narayan Mandayam, “Overview of OFDM, ”http://www.winlab.rutgers.edu/~nara yan/Course/WSID/wsid-lec1f.PPT.