ω ω - -2 m 0 ω -2 0 ω ω -2 + m 0 ω ω - - 0 m ω - 0 ω ω + - 0 m ω - m 0 ω m ω ω - m 0 ω 0 ω ω + m 0 ω ω - 2 m 0 ω 2 0 ω ω + 2 m 0

1

x

_c

(t)

p(t)

t

t

x

_s

(t)

t

-5T

₀

-4T

₀

-3T

₀

-2T

₀

-T0 0 T

₀

2T

₀

3T

₀

4T

₀

5T

₀

x

_c

(t)

p(t)

x

_s

(t)

SAMPLING AND RECONSTRUCTION OF SIGNALS

The sampling theorem states that to reconstruct any analog signal from its samples, the sampling frequency ω

₀

must be at least twice the signal’s maximum frequency ω

_m

:

ω

₀

≥ 2ω

_m

Sampling is a presentation of the continuous-time signal x

_c

(t) by a series of samples x[nT

₀

].

Consider a signal x

_s

(t) defined as the product of two signals: x

_c

(t)-is an original signal and p(t) -is a periodic

impulse train or Dirac distribution.

The spectrum of the Dirac distribution p(t) is itself a periodic train.

The spectrum X s( ω ) of the output signal x s(t)

The spectrum of the signals are shown in Figure 5.2; where Xc ω) is arbitrary. The distance between two adjacent replicated spectra is called a guard band

B g = ω0- 2 ωm.

∑

+∞ ∞ − ω − ω δ π = ω ( k ) T 2 ) ( P 0 0

∑

+∞ −∞ = ω − ω = ω ω π = ω k 0 c 0 c s _T X ( k ) 1 )] ( P * ) ( X [ 2 1 ) ( X (

∑

+∞

∑

−∞ = +∞ −∞ =

−

δ

=

−

δ

=

=

k k 0 0 c 0 c c s

(

t

)

x

(

t

)

p

(

t

)

x

(

t

)

(

t

kT

)

x

(

kT

)

(

t

kT

)

x

x

_s

(t)= x

_c

(t)p(t)

_∑

+∞ −∞ =

−

δ

=

k 0

)

kT

t

(

)

t

(

p

3

Xc( ) P( ω ) ω ω - 2 ω0 - ω0 0 ω0 2 ω0 Figure 5.2 Xs( ω ) ω -2 ω-ω0 m -2 ω0 -2 ω+0 ωm -ω0 -ωm -ω0 -ω0 + ωm _-ωm 0 ωm ω-ω0 m _ω0 ω+0 ωm 2 ω-ω0 m 2 ω0 2 ω+0 ωm B g = ω0- 2 ωm ωc c Xs( ω ) ω - 2 ω0 ω0 0 ω0 2 ω0 Aliasing - ωm ωm 1 2 π /T0 1/T 0 1/T 0 ω Xc( ω ) ₁ T0 ω m H(j ϖ ) - ωc a) b) c) d) e) f) ω

Example 5.1 A band limited signal has a bandwidth equal to 200 Hz. What sampling rate should be used to guarantee a guard band

of 100 Hz.

Solution: Fm=200 Hz; Bg=100 Hz. Bg= F0-2Fm; 100=F0-2x100; F0=300 Hz.

The following three cases present practical interest:

Under sampling:

ω0 < 2ωm

Nyquist rate:

ω0 = 2ωm

Over sampling:

ω0 > 2ωm

From Figure 5.2 it is evident that when ω0-ωm>ωm or ω0>2ωm the spectrum of Xs(ω) don’t overlap (see Figure 5.2 (c)). and

consequently it can be recovered from its samples with ideal low-pass filter having a frequency response H(jω) (see Figure 5.2 (e)).

If ω0>2ωm , output of the filter corresponds to Xc(ω) (see Figure 5.2 (f)). If ω0>2ωm does not hold , i.e ω0<2ωm the spectrum

Xs(ω) overlap(see Figure 5.2 (d)) and xc(t) is not recoverable by low pass filtering because of side-band distortion. This high

frequency distortion is called an aliasing.

5

Reconstruction of a Bandlimited Signal From Its Samples

According to the sampling theorem, samples of a continuous-time band limited signal taken frequently enough are sufficient to represent the signal exactly in the sense that the signal can be recovered from the samples. Impulse train modulation provides a convenient means for understanding the process of reconstructing the continuous-time bandlimited signal from its samples.

If the conditions of the sampling theorem are met and if the modulated impulse train is filtered by an appropriate low-pass filter, then the Fourier transform of the filter output will be

identical to the Fourier transform of the original continuous-time signal xc(t), and thus the

output of the filter will be x*c (t). If x c(nT0) is the input to an ideal low-pass continuous time

filter with frequency response Hr(jω) and impulse response hr(t), them the output of the filter

will be

[ ] [

r 0

]

n 0 c c (t) x nT h t nT x∗ =

∑

∞ − −∞ = (5.1)

A block diagram representation of this signal reconstruction process is shown in Figure 5.3.

x

c

[nT

0

]

_x*

_c

_(t)

x

c

(t)

Sampling

period T

0

Ideal

reconstruction

filter

H

r

(jω)

Convert from

sequence to

impulse train

Figure 5.3

[ ]

0 0 t j t j 0 t j 0 t j 0 r

T

/

t

T

/

t

sin

j

2

e

e

t

T

e

jt

1

2

T

d

e

2

T

)

t

(

h

c c c c c c

π

π

=

+

π

=

=

π

=

ω

π

=

ω ω ω ω − ω ω ω − ω

∫

T0 if |ω| < ωc

0 if |ω| > ωc

H(ω) =

7

» sxms t k

»x1=sin(2*(t -(4*k*pi/2)))/(2*(t -(4*k*pi/2))); z1=sxmsum(x1,k, -2,2); » x2=sin(2*(t -(4*k+1)*pi/2))/(2*(t -(4*k+1)*pi/2)); z2=sxmsum(x2,k, -2,2); » x3=sin(2*(t -(4*k+2)*pi/2))/(2*(t -(4*k+2)*pi/2)); z3=sxmsum(x3,k, -2,2); » x3=sin(2*(t -(4*k+3)*pi/2))/(2 *(t-(4*k+3)*pi/2)); z4=sxmsum(x3,k, -2,2); » x3=sin(2*(t -(4*k+4)*pi/2))/(2*(t -(4*k+4)*pi/2)); z5=sxmsum(x3,k, -2,2); » x*c=2.2975*z1 -0.5975*z2+0.2975*z3 -0.5975*z4; ezplot(x*c,[0 2*pi]) » t=0:2*pi/40:2*pi; » s=0.35+sin(t+pi/2)+ 1.34*sin(2*t+pi/4); » plot(t,s) »hold on » t=0:2*pi/4:2*pi; » s=0.35+sin(t+pi/2)+ 1.34*sin(2*t+pi/4); » stem(t,s,'fill','k') 0.2975 -0.5975 -0.5975 0 1 2 3 4 5 6 7 -0.5 -1.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 2.2975 _2.2975 0.2975 -0.5975 -0.5975 0 1 2 3 4 5 6 7 -0.5 -1.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 2.2975 _2.2975 x*c(t) xc(t)

A major application of discrete-time systems is the processing of continuous-time signals.

This is accomplished by a system of the general form depicted in Figure 5.8

ADC- Analog-to-digital converter; DAC- Digital –to-analog converter

x

c

(t)

X[n]

Y

[n]

y(t)

T

0

T

0

ADC

Discrete-time

system

DAC

Sampling Interval and Lagrange Approximation

Interpolation means to estimate a missing function value by taking weighted average values at

neighboring points.

The general form of Lagrange approximation passing true N+1 points

(

t

0

,

x

0

),..(

t

n

,

x

n

)

is

defined as

9

The Lagrange polynomial passing true the 2 points (t

1

,x

1

) and (t

2

,x

2

) is linear interpolation

∑

=

=

=

1 0 K K , 1 K 1

(

x

)

x

L

(

t

)

P

=

x

0

L

1,0

(

t

)

+

x

1

L

1,1

(

t

)

(5.5)

1 0 1 0 , 1

t

t

t

t

)

t

(

L

−

−

=

;

0 1 0 1 , 1

t

t

t

t

)

t

(

L

−

−

=

The Lagrange parabolic interpolating polynomial passing trough 3 points

(

t

0

,

x

0

),

(

t

1

,

x

1

)

and

)

,

(

t

2

x

2

is

∑

=

=

2 0 K K 2,K 2

(

t

)

x

L

(

t

)

P

) t t )( t t ( ) t t )( t t ( ) t ( L 2 0 1 0 2 1 0 , 2 _{− −} − − = ) )( ( ) )( ( ) ( 2 1 0 1 2 0 1 , 2 _{t t t t} t t t t t L − − − − = ) )( ( ) )( ( ) ( 1 2 0 2 1 0 2 , 2 _{t t t t} t t t t t L − − − − = For equally spaced nodes with t

t

0=0, t1=1 and t2=2

2 ) 2 )( 1 ( ) ( 0 , 2 − − = t t t L ; L_2,1(t) = −t(t−2); 2 ) 1 t ( t ) t ( L_2,2 = − 2 ) 1 t ( t y ) -2 t ( t y 2 ) 2 t -)( 1 t-( y ) t ( P_{2 = 0} − ₁ + ₂ −

Error of approximation ) t ( P ) t ( x ) t ( = _c − _N,K ε

Sampling intervals T 0for N=0; 1; 2 are defined as following.

N=0 – staircase approximation see (Figure 5.13)

1 0

M

T = ε ; (5.7)

N=1 – Linear interpolation (see Figure 5.14)

2 1 _M 8 T = ε ; M =₂ x ''(t) (5.8) N=2 – Parabolic interpolation 3 3 2 M 6 , 15 T = ;M₃ = x' ''(t) (5.9) t x(t) 0 T 2T 3T 4T t x(t) 0 T 2T 3T 4T x(t) Figure 5.14 Figure 5.13

11

Example 5.4. Using the Matlab files perform staircase and linear approximation of y=sin(t)

for t = 0: 2π.

In Figure 5.16 are shown staircase (a) and linear (b) interpolations of the sinusoidal signal using the Matlab files.

» t=0:pi/100:2*pi; y=sin(t); plot(t,y,'k'); hold on » t=0:pi/4:2*pi; y=sin(t); stem(t,y,'k','fill'); hold on

» stairs(t,y) set(gca,'xtick',[0.pi/4 pi/2 3pi/4 pi 5pi/4 6pi/4 7pi/4 2pi]) »t=0:2*pi/100:2*pi; y=sin(t);ti=0:pi/4:2*pi; yi=interp1(t,y,ti);plot(t,y,ti,yi)

a) b) 1 0 0.78 1.57 2.35 3.14 3.92 4.71 5.49 6.28 -0.8 -0.6 -0.4 -0.2 0 0.2 0.6 0.8 1 0 0.78 1.57 2.35 3.14 3.92 4.71 5.49 6.28 -0.8 -0.6 0.4 -0.2 Figure 5.16 0.8 0.4 0.6 0.4 0.2