Instructor : Dr. Phan Van CaLecture 05: Pulse Shaping

, Bandwidth Efficiencyand Demodulation

Digital Communications

A New Way of Viewing Modulation

The I/Q representation of modulation is very convenient for some modulation types.

We examine an even more general way of looking at modulation using signal spaces.

By choosing an appropriate set of axis for our signal constellation, we will be able to:

Design modulation types which have desirable propertiesConstruct optimal receivers for a given type of modulationAnalyze the performance of modulation types using very general techniques.

Summary of Gram-Schmidt Procedure

1st basis function is normalized version of 1st signal.

Successive basis functions are found by removing portions of signals which are correlated to previous basis functions, and normalizing the result.

This procedure is repeated until all basis functions are exhausted.

If , then no new basis function is added.

The order in which signals are considered is arbitrary.

f tk′ =( ) 0

Notes on Gram-Schmidt Procedure

A signal set may have many different sets of basis functions.

A change of basis functions is equivalent to rotating coordinates.

The order in which signals are used in the Gram-Schmidt procedure will affect the resulting basis functions.

The choice of basis functions does not effect performance.

Pulse Shaping - Why Does it Matter

One way of reducing bandwidth requirements is through efficient quantization

Sample rate: samples/second.

Bit rate out of the quantizer :

Once these two factors are determined, the bandwidth is given by :

The constant depends on the pulse shape

Example: first null bandwidth (with rectangular pulse shaping):

f sf M f ns slog2 = ⋅ bits/ second

BW Hz= ⋅ ⋅C f nPS s

f ns ⋅ Hz

CPS

Definitions of Bandwidth for Baseband Signals

Absolute Bandwidth

X dB Bandwidth

Y % Power Bandwidth

First Null Bandwidth

( )W f f B= >0, for

( ){ }( )

10 10

2log

max,

W f

W fX f B

⎡

⎣⎢⎢

⎤

⎦⎥⎥> > dB

( )( )

W f df

W f df

YBB 2

2 100−

−∞∞∫

∫≥

Design Criteria for Pulse Shapes

Two important characteristicsFirst null bandwidthSize of sidelobes

Would like to “round off corners” of pulses

f (Hz)

|W(f)|^2

BW

dB down

Rectangular Pulse

First Null BW: 1/T = 1 MHzFirst Sidelobe: 13.6 dB down

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (us)

x(t)

Time Domain Plot of 1 us Rectangular Pulse

-5 0 5-40

-30

-20

-10

0

10

20

f (MHz)

)|^2 (dB)

Energy Spectrum of 1 us Rectangular Pulse

Triangular Pulse

First Null BW: 2/T = 2 MHzFirst Sidelobe: 26 dB down

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (us)

x(t)

Time Domain Plot of 1 us Triangular Pulse

-10 -5 0 5 10-100

-80

-60

-40

-20

0

20

f (MHz)

|X(f)|^2 (dB)

Energy Spectrum of 1 us Triangular Pulse

Sinusoidal Pulse Shape

First Null BW: 1.5/T = 1.5 MhzFirst Sidelobe: 22 dB down

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (us)

x(t)

Time Domain Plot of 1 us Sinusoidal Pulse

-10 -5 0 5 10-80

-60

-40

-20

0

20

40

f (MHz)

|X(f)|^2 (dB)

Energy Spectrum of 1 us Sinusoidal Pulse

Truncated Gaussian Pulse Shape

First Null BW: 1.5/T = 1.5 MHzLargest Sidelobe: 31 dB down

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (us)

x(t)

Time Domain Plot of 1 us Truncated Gaussian Pulse

-10 -5 0 5 10-70

-60

-50

-40

-30

-20

-10

0

10

20

30

f (MHz)

|X(f)|^2 (dB)

Energy Spectrum of 1 us Truncated Gaussian Pulse

Intersymbol Interference

Though we may refine our pulse shape further, it is clear that we are close to the limit.

One way to achieve a narrow spectral is to use longer duration pulses.

If pulses overlap, they may produce intersymbolinterference

time

w(t)

Nyquist’s First Criteria for Zero ISI

Overlapping pulses will not cause intersymbol interference if they have zero amplitude at the time we sample the signal.

Mathmatically:

where k is an integer and is one symbol duration

( )h kTC k

ke s ==≠

⎧⎨⎩

,

,

0

0 0

Ts

Raised Cosine Pulse Family -Satisfies the Nyquist Criteria

Frequency Domain:

B is the absolute bandwidth of the filter

Time Domain:

( ) ( )H f

f ff f

ff f B

f B

e =

<

+−⎛

⎝⎜⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ ≤ ≤

>

⎧

⎨⎪⎪

⎩⎪⎪

1

12

120

1

11

,

cos ,

,

π

∆

0 1 0 0, ,f B f f f f r f f∆ ∆ ∆= − = − =

( ) ( ){ } ( ) ( )( )

h t F H f ff t

f t

f t

f te e= =

⎛⎝⎜

⎞⎠⎟⋅

−

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−10

0

0 22

2

2

2

1 4

sin cosππ

π ∆

∆

Spectrum of Raised Cosine Pulse

r=0 corresponds to Sa(.) function

f (Hz)

1.0

0.5

f1 f0 B

f∆ f∆

( )H fe

Raised Cosine Pulse - Time Domain

r=0r=0.35r=0.5r=1.0

-4 -3 -2 -1 0 1 2 3 4-0.5

0

0.5

1

1.5

2

t (us)

x(t)

Raised Cosine Pulses for Several Different Rolloff Factors

Raised Cosine Pulse - Frequency Domain

r=0r=0.35r=0.5r=1.0

-5 0 5-120

-100

-80

-60

-40

-20

0

20

40

f (MHz)

|X(f)|^2 (dB)

Energy Spectrum of Raised Cosine Pulses

Implementation of Raised Cosine Pulse

Can be digitally implemented with an FIR filter

Analog filters such as Butterworth filters may approximate the tight shape of this spectrum

Practical pulses must be truncated in timeTruncation leads to sidelobes - even in RC pulses

Sometimes a “square-root” raised cosine spectrum is used when identical filters are implemented at transmitter and receiver

We will discuss this more when we talk about “matched filtering”

Truncated Raised Cosine Pulses

Truncating raised cosine pulse to finite duration results in some sidelobes

r=0r=0.35r=0.5r=1.0

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

0

0.5

1

1.5

2

t (us)

x(t)

Truncated Raised Cosine Pulses for Different Rolloff Factors

r=0r=0.35r=0.5r=1.0

-5 0 5-100

-80

-60

-40

-20

0

20

40

f (MHz)

|X(f)|^2 (dB)

Energy Spectrum of Truncated Raised Cosine Pulses

Raised Cosine (RRaised Cosine (R--C) Filter ( 2 C) Filter ( 2 -- ASK , r = 0 )ASK , r = 0 )

Roll-off Factor : r = 0

Filter Duration = [-4Ts,4Ts]

No. of Oversampling = 8

Output of Matched Filter

[1,-1,-1,1,-1,1,1,-1,1,-1]

Raised Cosine Filter ( 2 Raised Cosine Filter ( 2 -- ASK , r = 0.5 )ASK , r = 0.5 )

Roll-off Factor : r = 0.5

Filter Duration = [-4Ts,4Ts]

No. of Oversampling = 8

Output of Matched Filter

[1,-1,-1,1,-1,1,1,-1,1,-1]

Raised Cosine Filter ( 2 Raised Cosine Filter ( 2 -- ASK , r = 1 )ASK , r = 1 )

Roll-off Factor : r = 1.0

Filter Duration = [-4Ts,4Ts]

No. of Oversampling = 8

Bandwidth Considerations

Signal bandwidth is determined from transmitted signal’s power spectral density .

Computation of is discussed in Section 4-4-1, We summarize important results.

We define bandwidth as the range of positive frequency for which signal is non-negligible. That may mean:

First spectral null occurs within a bandwidth99% of power is contained with a bandwidthAll spectral components are 40 dB down from peak value

( )Φss f

( )Φss f

Factors Affecting Bandwidth

Symbol Rate (related to bit rate by )

- large implies short pulses and large BW

Number of dimensions K of signal space.

Ways of implementing dimensions:Distinct Time Slots

Distinct Frequency Bands

Signals in Quadrature

Type of pulse shaping employed

Rs R R Mb s= log2Rs

Pulse Shaping

Rectangular pulses have first null BWBW = (Baseband signal)

BW = (Bandpass signal)

Optimum pulse shape has absolute BWBW = (Baseband signal)

BW = (Bandpass signal)

How is optimum pulse shape implemented?Raised Cosine Pulse

2Rs

Rs / 2Rs

Rs

Bandwidth Requirements for Bandpass Modulation (Assuming Both I&Q Used)

Optimum Pulse Shaping:

Rectangular Pulse Shaping (a good rule of thumb):

Raised Cosine Pulse Shaping:

If quadrature channel is not used (e.g. BPSK), then BW is twice as large.

BWR

KR

MKs b= ⋅ = ⋅

2 2 2log

BW R K KR Ms b= ⋅ = / log2

( ) ( )BWR

K rR

MK rs b= ⋅ ⋅ + = ⋅ ⋅ +

21

21

2log

Bandwidth Efficiency:

Definition: (bits/sec)/Hz

Measures how efficiently a modulation type uses bandwidth

Typical Values (assuming optimum pulse shaping):BPSK: 1 bits/sec/HzQPSK: 2 bits/sec/Hz8-ary PSK: 3 bits/sec/Hz16 QAM: 4 bits/sec/Hz2-ary FSK: 0.5 bits/sec/Hz8-ary FSK: 3/8 bits/sec/Hz

ηB bR= / BW

Modulation

We want to modulate digital data using signal sets which are:

bandwidth efficient

energy efficient

A signal space representation is a convenient form for viewing modulation which allows us to:

design energy and bandwidth efficient signal constellations

determine the form of the optimal receiver for a given constellation

evaluate the performance of a modulation type

Problem Statement

We transmit a signal , where

is nonzero only on .

Let the various signals be transmitted with probability:

The received signal is corrupted by noise:

Given , the receiver forms an estimate of the signal with the goal of minimizing symbol error probability

( ) ( ){ }s t s t s t s tM∈ 1 2, ( ), , ( )…( )s t [ ]t T∈ 0,

( )[ ] ( )[ ]p s t p s tM M1 1= =Pr , , Pr…

( ) ( ) ( )r t s t n t= +r t( ) ( )s t

( )s t

( ) ( )[ ]P s t s ts= ≠Pr

Noise Model

The signal is corrupted by Additive White Gaussian Noise (AWGN)

The noise has autocorrelation and power spectral density

Any linear function of will be a Gaussian random variable

n t( )

n t( )

( ) ( )φ τ δ τnnN

= 02( )Φnn f N= 0 2

n t( )

s t( )

n t( )

r t( )Σ

Channel

Signal Space Representation

The transmitted signal can be represented as:

,

where .

The noise can be respresented as:

where

and

( )s t s f tm m k kk

K( ) ,= ∑

=1

( )s s t f t dtm k m kT

, ( )= ∫0

( ) ( )n t n t n f tk kk

K= ′ + ∑

=( )

1

( ) ( )n n t f t dtk kT

= ∫0

( ) ( )′ = − ∑=

n t n t n f tk kk

K( )

1

Signal Space Representation (continued)

The received signal can be represented as:

where

( ) ( ) ( ) ( )r t s f t n f t n t r f t n tm k kk

Kk k

k

Kk k

k

K= ∑ + ∑ + ′ = + ′∑

= = =, ( ) ( )

1 1 1

r s nk m k k= +,

The Orthogonal Noise:

The noise can be disregarded by the receiver

( ) ( )

( ) ( )

( ) ( )

s t n t dt s t n t n f t dt

s f t n t n f t dt

s f t n t dt s n f t dt

s n s n

mT

m k kk

KT

m k kk

Kk k

k

KT

m kk

Kk

Tm k k

k

K

k

T

m k kk

Km k k

k

K

( ) ( ) ( )

( )

( )

,

, ,

, ,

′∫ = − ∑⎛

⎝⎜

⎞

⎠⎟∫

= ∑ − ∑⎛

⎝⎜

⎞

⎠⎟∫

= ∑ ∫ − ∑ ∫

= ∑ − ∑ =

=

= =

= =

= =

0 10

1 10

1 0 1

2

0

1 10

( )′n t

( )′n t

We can reduce the decision to a finite dimensional space!

We transmit a K dimensional signal vector:

We receive a vector which is the sum of the signal vector and noise vector

Given , we wish to form an estimate of the transmitted signal vector which minimizes

[ ] { }s s s= ∈s s sK M1 2 1, , , , ,… …[ ]r s n= = +r rK1, ,…

[ ]n= n nK1, ,…

r s[ ]Ps = ≠Pr s s

s Σ

Channel

n

r Receiver s

MAP (Maximum a posteriori probability) Decision Rule

Suppose that signal vectors are transmitted with probabilities respectively, and the signal vector is received

We minimize symbol error probability by choosing the signal which satisfies:

Equivalently:

or

{ }s s1, ,… M{ }p pm1, ,…

r

sm ( ) ( )Pr Pr ,s r s rm i m i≥ ∀ ≠

( ) ( )( )

( ) ( )( )

p

p

p

pm im m i ir s s

r

r s s

r

Pr Pr,≥ ∀ ≠

( ) ( ) ( ) ( )p p m im m i ir s s r s sPr Pr ,≥ ∀ ≠

Maximum Likelihood (ML) Decision Rule

If or the a priori probabilities are unknown, then the MAP rule simplifies to the ML Rule

We minimize symbol error probability by choosing the signal which satisfies

p pm1 = =

sm ( ) ( )p p m im ir s r s≥ ∀ ≠,