Causal Linear Time-Invariant systemsLTI system is causal if and only if its impulse response is zero for negative values of n.For a causal system, h(n)=0 for n<0, the limits on the summation of the convolution formula is given as

y(n)= h(k)x(n-k) ………….(2.29a)

= x(k) y(n-k)………(2.29b)

Causality is required in any real time signal processing applications, since at any given time n we have no access to future values of the input signal. Only the present and past values and available to determine the present output.

Stability of Linear Time Invariant systemsA LTI system is stable if its impulse response is absolutely assumable. This condition is necessary to ensure the stability of the system.

For instance, if Sn= , here is a bomeded input for which the o/p is not bounded, if we choose the bomnded input as

S(n)=

were h*(n) is the complex conjugate of h(n). It is sufficient to show that there is one value of n for which y(n) is unbounded. For n=0, we have,

y(o)=

Now, if Sh=, a bounded input produces an unbounded output since y(0)= . The output of a system is bounded if the impulse response of the system satisfies the condition.

Sh …….. (2.30)

This implies that the impulse response h (n) goes to zero as n approaches infinity. Thus, the o/p of the system goes to zero as n approaches infinity if the input is set to zero beyond n> n0.

AssignmentDetermine the range of values of a and b for which the linear time invariant system with impulse response.

h(n)=is stable. Implementation of discrete-Time systems.

System design and implementation are usually goes together. The system design is driven by the method of implementation and implementation constraints like cost, Hardware limitations, size limitations, and power requirements. Now, we will just look at the basic implementation method of LTI system. System analysis, Design and implementation will be treated in details hater.

2.4.1 Structures for the Realization of Linear Time-Invariant system. The structures for the realization of systems described by linear constant- coefficient difference equations as shown in fig 2.3 considering the 1st order system given by

y(n)= -ay(n-1)+ b0x(n)+b1x(n-1)…..(2.310It uses separate delays (memory) for both the input and output signal samples and it is called a direct form I structure. The system can be viewed as two liner time-invariant systems in cascade. The first is a non recursive given by

V(n)= box(n) + b1x(n-1)…….(2.32)and the second is a recursive system described by the equation.

y(n)= -a1y(n-1)+ v(n)…….(2.32)Thus, if we interchange the order of the cascaded linear time-invariant systems the over all system response remains the same. So if we interchange the order of the recursive and non-recursive systems, it gives as an alternative structure to generate the system described in equ. 2.31. Fig 2.3(b) is the resulting system which gives two difference equations as

w(n) = - a1 W(n-1)+x(n)…….(2.34)y(n) = b0W(n) + b1W(n-1)……(2.35)

Hence, the two difference equation (2.34) and (2.35) are equivalent to the single difference equation (2.31)

In fig 2.3, the two delay elements contain the same input w(n) and the same output w(n-1). Thus, the two elements can be merged into one delay as shown in fig 2.3C. This is more efficient in terms of memory requirements and it is known as direct form // structure and it is used in practical applications.

In general, linear time-invariant recursive system described by the deference equation is given as

y(n)= - ……(2.36)

the direct form 1 structure for this system is shown in fig 2.4. This structure requires M+N delays and N+ M+ 1 multiplication. The cascade of a non-recursive system is

V(n)= …… (2.37)

and a recursive system is

y(n)= …..(2.38)

By reversing the order of this two systems, we obtain the direct Form //Structure as shown in fig 2.5for N>m. This structure is the cascade of a recursive system

W(n)= - …….(2.39

and followed by a non recursive system

y(n)= … …(2.40)

Z-1 Z-1

++

b1 -a1 (a)

y(n)x(n) b0 V(n)

Fig. 2.3 steps in converting from the direct form I realization in (a)to the direct form II realization in(c)

Z-1 Z-1

++b0

-a1

(b)b1

y(n)w(n)x(n)

w(n-1)w(n-1)

Z-1

++ b0

-a1

( c)

b1

y(n)w(n)x(n)

w(n-1)

Z-1

++x(n)

Z-1

Z-1

+

+

+

b1

b2

bm-1

bm

b0 y(n)

Z-1

+

+

Z-1

Z-1

+

+

-a1

-a2

-aN-1

-a4

V(n)

Fig 2.4. Direct Form I structure of the system described by equ 2.36.

Z-1

Z-1

Z-1

Z-1

Z-1

+

+

+

+

+

+

+

+

+

b0

b1

b2

b3

bm

-a1

-a2

-a3

-aN-2

-aN

-aN-1

(m=N-2)

y(n)x(n)

If ak=0 k=1,2……,N. Then, the input output relation ship for the system in equ 2.36 reduces to

y(n)= ……(2.41)

Which is a non recursive linear time invariant system. In this case, the system output is basically a weighed moving average of the input signal; hence, it is also called a moving average (mA) system. Thus, such a system is an FIR system with an impulse response h(k) equal to the co-efficient bk, ie.

h(k)= ….(2.42)But, it equ 2.36, we set m=0 , then, the general linear time-invariant system reduces to a “purely recursive” system described by the difference equation.

y(n) = - …..(2.43)

Thus, the system output is a weighted linear combination of N past outputs and the present input.

The general second-order system is described by the difference equation.

y(n)= - a1y(n-1) –a2y(n-2)+box(n)+b1x(n-1)+b2x(n-2)……(2.44)This is obtained from equ 2.36 by settp N= 2 and m=2. The direct form II structure for realizing this system is shown in fig 2.6a. if weset aa=a2=0, then equ. 2.44 reduces to

y(n) = b0x(n)+b1x(n+1)+b2x(n-2)…..2.45this is a special case of the FIR system described by equ. 2.41as shown in fig 2.66 . Also , if reset b1=b2=0, the equation reduces to

y(n)= - a1y(n-1)-a2y(n-2)+b0x(n)…..(2.46)which is a special case of pare recursive system and it is shown in fig 2.6c

fig 2.6. Structures for the realization of second order systems. (a) General second –order system; (b) FIR system; (c) purely recursive system.

Recursive Nonre cursive Realizations of FIR System is described by am input-Output equation of the form

Z-1

Z-1

+

+

+

+

b0

b1

b2

-a1

-a2

y(n)x(n)+

(a)

Z-1

++

Z-1

b1 b2b0

x(n)

(b)

y(n)

-a1

x(n)

(c)

Z-1

+ +

Z-1

-a2

y(n)=F[y(n-1), ….., y(n-N), x(n),….,x(n-m)….(2.47),and for alinear time –invariant system specifically, by the difference equation

y(n)= - ……(2.48)

white, causal non recursive systems do not depend on past values of Output and hence are described by an input-output equation of the form

y(n)=F(x(n),s(n-1),..,x(n-M)…..(2.49)and for linear time-invariant specifically, by the difference equation in (2.48) with ak=0 for K=1,2,…NIn the case of F/R Systems, we have a system with an input –output equation.

y(n)= ……..(2.50)

This is a non recursive and F/R system.

2.5 Correlation of Discrete-Time Signals. Correlation resembles convolution. Kike in Convolution, two signal sequences are involved in correlation. In correlation out objectives is to measure the degree to which the two synals are similar and thus to extract some information that depends to a large extent on the applications. Correclation of signals is often concomtered in radar, sonar, digital communications, geology etc.

Suppose we have two signal sequences x(n) and that we wish to compare. In radar and active souar applications, x(n) represent the sampled version of the transmitted signal and y(n) represent the sampled version of the received signal at the output of the (A/D) Converter. If a target is presenting the space benig searched by the radar or souar, the received signal y(n) consists of a delayed version of the transmitted signal, reflected from the target, and correpted by additive noise as shown in fig 2.7. The received signal sequence is represented as

y(n)= x(n-D)+W(n)……(2.51)Where is some attenuation factor representing the signal loss involved in the round-trip transmission of the signal x(n), D the round-trip delay and w(n) represents the additive noise that is picked up by the antenna and any noise generated by the electronic components and amplifiers contained in the front end of the receiver.

But , if there is no target in the space searched by the radar and sonar, the received signal y(n) consists of noise alone.

From the two signal sequences, x(n), which is called the reference signal or transmitted signal, and y(n),the received signal , the problem in radar and Fonar detection is to compare y(n) and x(n) to determine if a target is present and if so , to determine the time delay D and compnter the distance to the target. In practvce, the signal x(n-d) is hearily corrupted by the additive noise to the point where a risual inspection of y(n) does not reveal the presenceor absence of the desired signal reflected from the target. Thus, correlation providesus with a means for extracting this important information from y(n).

Digital communications is another area where conrrelation is often used. Hare, the information to be transmitted from one point to another is usually converted to binary

form, it, a sequence of zeros and is which are then transmitted to the intended receiver. To transmit a 0 we can transmit the signal sequence X0(n) for 0 ≤ n≤ L-1 and to transmit a 1 we can transmit the signal sequence x1(n) for 0 ≤ n≤ L-1, where L is an integer that denotes the number of samples in each of the two sequences. The signal received by the intended receiver may be represented as

y(n) =Xi(n) + W(n) i=0,1 0 ≤ n≤< -1 …..(2.52)where now the uncertainty is whether X0(n) or X1(n) is the Signal component in y(n), and W(n) represents the additive noise and other interference inherent in any communication system.

Cross correlation and Autocorrelation Sequences If the two real signal sequences X(n) and y(n) each of which has finite energy. Thus, the cross correlation of X(n) and y(n) is a sequence rxy(L), which is defined as

rxy(L)= L=0,1, 2, …….(2.53)

equivalent as

rxy(L) = l= 0, 2, ……(2.54)

where, L = is the (time) shift(or lag) parameter Xy= is the subscript on the cross correlation correlated sequencerxy (i) indicates the

sequences being corrdated The order of the subscripts, with x preceding y, indicates the direction in which one sequence is shifted, relative to the othe. This implies that as shown in equ.2.53,thesequence x(n) is left un shifted an y(n) is shifted by I units in time, to the right for l negative. Also, in equ 2.54, the sequence y(n) is left un shifted and x(n)is shifted by (un it sin time , to the left for l positive and to the right for (negative. But shifting x(n) to the left by l units relative to y(n) is equivalent to shift to shafting y(n) to the right by l units relative to x(n). Hence, the above two equations field identical cons correlation sequences.

ryx(l)= …….(2.55)

or

rgx(l)= ……(2.56)

By comparing (2.53) with (2.56) or (2.54) with (2.55), we conclude that rxr(l)= ryx(-1)……(2.57)

is to compare y(n) and x(n) to determine if a target is present and ,if so, to determine the time delay D and compnter the distance to the target. In practice, the signal x(n-D) is hearily corrupted by the additive noise to the point where a risual inspection of y(n) does not reveal the presence or absence of the desired signal reflected from the target. Thus, correlation provides us with a means for extracting this important information from y(n)

Digital communications is another area where correlation is of ten used. Hare, the information to be transmitted from one point to another is usually, Commented to binary form, ie. as equence of xeros and is which are then transmitted to the intended receiver. To transmit a0 we can transmit the signal sequence x0(n) for 0≤n≤L -1 , where L is an integer that denotes the number of samples weach of the two sequences. The signal received by the intended receiver may be represented as

y(n)=Xi(n)+ w(n) i=1,1 0 ≤ n ≤ L-1….(2.52)where now the uncertainty is whether X0(n) or X1(n) is the signal component in y(n), and W(n) represents the additive noise and other interference inherent in any communication system.

Cross correlation and Auto correlation Sequences If the two real Signal Sequences X(n) and y(n) each of which has finite energy. Thus, the cross correlation of x(n) and y(n) is a sequence rxy(l) , which is defined as

rxy(l) = L= 0, 1, 2, ………..(2.53)

Example: Determine the cross correlation sequence rxy(l) of the sequencesx(n) ={..., 0,0,2,-1,3,7,1,2,-3,0,0, …}y(n)= { …., 0,0,1,-1,2,-2,4,1,-2,5,0,0,…}Solution Applying equ 2.53, for l=0 , we have,

rxy(0) =

The product sequence V0(n) = x(n) y(n) isV0(n)= { …,0,0,2,1,6, -14,4,2,6,0,0,…}and the sum over all values of n is

rxy(0) =7For 1>0, we shift y(n) to the rish relative to x(n) by units Then, the product sequence

V1(n)= x(n) y(n-1) and sum over all values of the product sequence, we have.rxy(1) = 13, rxy(2)= -18, rxy(3)= 16, rxy(4)=-7 rxy(5)=5 , rxy(6)= -3, rxy(l) = 0 , l 7.

But, for l < 0, we shift y(n) to the left relative to x(n) by (unit compute the product sequence

V1(n)= x(n) y(n-l), and the sum over all values of the product sequence, hence, we obtain, rxy(-1) = 0, rxy(-2) = 33, rxy(-3)=-14, rxy(-14)= 36 rxy(-5)=19, rxy(-6)=-9, rxy(-7)=10, rxy(l)=0, l -8 the cross correlation sequence of x(n) and y(n) rxy(l)= {10,-9,19,36,-14,33,0,7,13,-18,16,-7,5,-3}Thus, except for the folding operation, the computation of the cross correlation sequence mvolves the same operations. i.e. - Shifting one of the sequences.- Multiplication of the two sequences.- Summing over all values of the product sequenceWhen y(n)= x(n), we have the auto correlation of x(n), which is defined as the sequence

rxy(l)= ……(2.58)

which is equivalent to

rxy(l)= …(2.59)

Correlation of periodic sequences

Previously, we considered energy single, but now we will look into the correlation sequences of power signals and in, particular, periodic signale.

Let x(n)= line …….(2.60)

mBut, if x(n)=y(n), we have the definition of the autocorrelation sequence of a power signal as

rxx(l) = line … ….(2.61)

m