In the previous post we have seen linear and non linear systems. Here we will see how to determine whether the system is stable or unstable i.e. stability property. To define stability of a system we will use the term ‘BIBO’. It stands for Bounded Input Bounded Output. The meaning of word ‘bounded’ is some finite value. So bounded input means input signal is having some finite value. i.e. input signal is not infinite. Similarly bounded output means, the output signal attains some finite value i.e. the output is not reaching to infinite level.

Definition of stable system:

An infinite system is BIBO stable if and only if every bounded input produces bounded output.

Mathematical representation:

Let us consider some finite number Mx whose value is less than infinite. That means Mx < ∞, so it’s a finite value. Then if input is bounded, we can write,

|x(n)| ≤ Mx < ∞

Similarly for C.T. system

|x(t)| ≤ Mx < ∞

Similarly consider some finite number My whose value is less than infinity. That means My < ∞, so it’s a finite value. Then if output is bounded, we can write,

|y(n)| ≤ My < ∞

Similarly for continuous time system

|y(t)| ≤ My < ∞

Definition of Unstable system:

An initially system is said to be unstable if bounded input produces unbounded (infinite) output.

Significance:

Unstable system shows erratic and extreme behavior.

When unstable system is practically implemented then it causes overflow.

Solved problem on stability:

Determine whether the following discrete time functions are stable or not.

1) y(n) = x(-n)

Solution: we have to check the stability of the system by applying bounded input. That means the value of x(-n) should be finite. So when input is bounded output will be bounded. Thus the given function is Stable system.

Hi friends, today we will learn What is Causal and non-causal system?. These two are very important system in control systems. These systems are distinguished from their input output relationship. Let us see these systems one by one.

A) Causal systems:

Definition: A system is said to be causal system if its output depends on present and past inputs only and not on future inputs.

Examples: The output of casual system depends on present and past inputs, it means y(n) is a function of x(n), x(n-1), x(n-2), x(n-3)…etc. Some examples of causal systems are given below:

1) y(n) = x(n) + x(n-2)

2) y(n) = x(n-1) – x(n-3)

3) y(n) = 7x(n-5)

Significance of causal systems:

Since causal system does not include future input samples; such system is practically realizable. That mean such system can be implemented practically. Generally all real time systems are causal systems; because in real time applications only present and past samples are present. Since future samples are not present; causal system is memory less system.

B) Anti causal or non-causal system:

Definition: A system whose present response depends on future values of the inputs is called as a non-causal system.

Examples: In this case, output y(n) is function of x(n), x(n-1), x(n-2)…etc. as well as it is function of x(n+1), x(n+2), x(n+3), … etc. following are some examples of non-causal systems:

1) Y(n) = x(n) + x(n+1)

2) Y(n) = 7x(n+2)

3) Y(n) = x(n) + 9x(n+5)

Significance of non-causal systems:

Since non-causal system contains future samples; a non-causal system is practically not realizable. That means in practical cases it is not possible to implement a non-causal system.

But if the signals are stored in the memory and at a later time they are used by a system then such signals are treated as advanced or future signal. Because such signals are already present, before the system has started its operation. In such cases it is possible to implement a non-causal system.

Some practical examples of non-causal systems are as follows:

1) Population growth

2) Weather forecasting

3) Planning commission etc.

For continuous time (C.T.) system:

A C.T. system is said to be “causal” if it produces a response y(t) only after the application of excitation x(t). That means for a causal system the response does not begin before the application of the input x(t).

The other way of defining the causal system is as follows:

A system is said to be “causal” if its output depends on present and past values of the input and not on the future inputs. If the input is applied at t = tm then the output at t = tm y(tm) will be dependent only on the values of x(t) for t ≤ tm.

Condition for causality: y(tm) = f[x(t); t ≤ tm]

Causal systems are physically realizable systems. The non-causal systems do not satisfy above condition. Non-causal systems are not physically realizable.

Condition for causality in terms of impulse response h(t):

The relation between y(t) and x(t) is given by,

y(t) = x(t)*h(t)

Where * represents convolution and h(t) is the impulse response of the system. The condition for causality in terms of the impulse response is as follows:

Condition for causality: h(t) = 0 for t<0

This condition states that a linear time invariant (LTI) system is “causal” if its impulse response h(t) has a zero value for negative values of time.

Solved problems on causal and non-causal system:

Determine if the systems described by the following equations are causal or non-causal.

1) y(n) = x(n) + x(n-3)

Solution: the given system is causal because its output (y(n)) depends only on the present x(n) and past x(n-3) inputs.

2) y(n) = x(-n+2)

Solution: this is non-causal system. This is because at n = -1 we get y(-1) = x[-(-1)+2] = x(3). Thus present output at n = -1, expects future input i.e. x(3)

I hope now u are able to understand the difference between causal and non-causal systems. If liked this post please share it with your friends and like our facebook page for future updates.

Linear or Non-linear Systems (Linearity Property):

A linear system is a system which follows the superposition principle. Let us consider a system having its response as ‘T’, input as x(n) and it produces output y(n). This is shown in figure below:

Let us consider two inputs. Input x1(n) produces output y1(n) and input x2(n) produces output y2(n). Now consider two arbitrary constants a1 and a2. Then simply multiply these constants with input x1(n) and x2(n) respectively. Thus a1x1(n) produces output a1y1(n) and a2x2(n) produces output a2y2(n).

Theorem for linearity of the system:

A system is said to be linear if the combined response of a1x1(n) and a2x2(n) is equal to the addition of the individual responses.

The above theorem is also known as superposition theorem.

Important Characteristic:

Linear system has one important characteristic: If the input to the system is zero then it produces zero output. If the given system produces some output (non-zero) at zero input then the system is said to be Non-linear system. If this condition is satisfied then apply the superposition theorem to determine whether the given system is linear or not?

For continuous time system:

Similar to the discrete time system a continuous time system is said to be linear if it follows the superposition theorem.

Let us consider two systems as follows:

y1(t) = f[x1(t)]

And y2(t) = f[x2(t)]

Here y1(t) and y2(t) are the responses of the system and x1(t) and x2(t) are the excitations. Then the system is said to be linear if it satisfies the following expression:

A system is said to be non-linear system if does not satisfies the above expression. Communication channels and filters are examples of linear systems.

How to determine whether the given system is Linear or not?

To determine whether the given system is Linear or not, we have to follow the following steps:

Step 1: Apply zero input and check the output. If the output is zero then the system is linear. If this step is satisfied then follow the remaining steps.

Step 2: Apply individual inputs to the system and determine corresponding outputs. Then add all outputs. Denote this addition by y’(n). This is the R.H.S. of the 1^{st} equation.

Step 3: Combine all inputs. Apply it to the system and find out y”(n). This is L.H.S. of equation (1).

Step 4: if y’(n) = y”(n) then the system is linear otherwise it is non-linear system.

Solved problem:

Determine whether the following system is linear or not?

y(n) = n x(n)

Solution:

Step 1: When input x(n) is zero then output is also zero. Here first step is satisfied so we will check remaining steps for linearity.

Step 2: Let us consider two inputs x1(n) and x2(n) be the two inputs which produces outputs y1(t) and y2(t) respectively. It is given as follows:

Now add these two output to get y’(n)

Therefore y’(n) = y1(n) + y2(n) = n x1(n) + n x2(n)

Therefore y’(n) = n [x1(n) + x2(n)]

Step 3: Now add x1(n) and x2(n) and apply this input to the system.

Therefore

We know that the function of system is to multiply input by ‘n’.

Here [x1(n) + x2(n)] acts as one input to the system. So the corresponding output is,

y”(n) = n [x1(n) + x2(n)]

Step 4: Compare y’(n) and y”(n).

Here y’(n) = y”(n). hence the given system is linear.

A system is said to be Time Invariant if its input output characteristics do not change with time. Otherwise it is said to be Time Variant system.

Explanation:

As already mentioned time invariant systems are those systems whose input output characteristics do not change with time shifting. Let us consider x(n) be the input to the system which produces output y(n) as shown in figure below.

Now delay input by k samples, it means our new input will become x(n-k). Now apply this delayed input x(n-k) to the same system as shown in figure below.

Now if the output of this system also delayed by k samples (i.e. if output is equal to y(n-k)) then this system is said to be Time invariant (or shift invariant) system.

If we observe carefully, x(n) is the initial input to the system which gives output y(n), if we delayed input by k samples output is also delayed by same (k) samples. Thus we can say that input output characteristics of the system do not change with time. Hence it is Time invariant system.

Theorem:

A system is Time Invariant if and only if

Similarly a continuous time system is Time Invariant if and only if

Now let us discuss about How to determine that the given system is Time invariant or not?

To determine whether the given system is Time Invariant or Time Variant, we have to follow the following steps:

Step 1: Delay the input x(n) by k samples i.e. x(n-k). Denote the corresponding output by y(n,k).

That means x(n-k) → y(n,k)

Step 2: In the given equation of system y(n) replace ‘n’ by ‘n-k’ throughout. Thus the output is y(n-k).

Step 3: If y(n,k) = y(n-k) then the system is time invariant (TIV) and if y(n,k) ≠ y(n-k) then system is time variant (TV).

Same steps are applicable for the continuous time systems.

Solved Problems:

1) Determine whether the following system is time invariant or not.

y(n) = x(n) – x(n-2)

Solution:

Step 1: Delay the input by ‘k’ samples and denote the output by y(n,k)

Therefore y(n,k) = x(n-k) – x(n-2-k)

Step 2: Replace ‘n’ by ‘n-k’ throughout the given equation.

Therefore y(n-k) = x(n-k) – x(n-k-2)

Step 3: Compare above two equations. Here y(n,k) = y(n-k). Thus the system is Time Invariant.

2) Determine whether the following systems are time invariant or not?

y(n) = x(n) + n x(n-2)

Solution:

Step 1: Delay the input by ‘k’ samples and denote the output by y(n,k)

Therefore y(n,k) = x(n-k) + n x(n-2)

Step 2: Replace ‘n’ by ‘n-k’ throughout the given equation.

Therefore y(n-k) = x(n-k) + (n-k) x(n-k-2)

Step 3: Compare above two equations. Here y(n,k) ≠ y(n-k). Thus the system is Time Variant.

Definition: It is a system in which output at any instant of time depends on input sample at the same time.

Example:

1) y(n) = 9x(n)

In this example 9 is constant which multiplies input x(n). But output at nth instant that means y(n) depends on the input at the same (nth) time instant x(n). So this is static system.

2) y(n) = x^{2}(n) + 8x(n) + 17

Here also output at n^{th} instant, y(n) depends on the input at nth instant. So this is static system.

Why static systems are memory less systems?

Answer:

Observe the input output relations of static system. Output does not depend on delayed [x(n-k)] or advanced [x(n+k)] input signals. It only depends on present input (nth) input signal. If output depends upon delayed input signals then such signals should be stored in memory to calculate the output at n^{th} instant. This is not required in static systems. Thus for static systems, memory is not required. Therefore static systems are memory less systems.

b) Dynamic systems:

Definition: It is a system in which output at any instant of time depends on input sample at the same time as well as at other times.

Here other time means, other than the present time instant. It may be past time or future time. Note that if x(n) represents input signal at present instant then,

1) x(n-k); that means delayed input signal is called as past signal.

2) x(n+k); that means advanced input signal is called as future signal.

Thus in dynamic systems, output depends on present input as well as past or future inputs.

Examples:

1) y(n) = x(n) + 6x(n-2)

Here output at n^{th} instant depends on input at n^{th} instant, x(n) as well as (n-2)^{th} instant x(n-2) is previous sample. So the system is dynamic.

2) y(n) = 4x(n+7) + x(n)

Here x(n+7) indicates advanced version of input sample that means it is future sample therefore this is dynamic system.

Why dynamic system has a memory?

Answer:

Observe input output relations of dynamic system. Since output depends on past or future input sample; we need a memory to store such samples. Thus dynamic system has a memory.

For continuous time (CT) systems:

A continuous time system is static or memoryless if its output depends upon the present input only.

Example:

Voltage drop across a resistor.

It is given by,

v(t) = i(t)*R

Here the voltage drop depends on the value of the current at that instant. So it is static system.

On the other hand a CT system is dynamic if output depends on present as well as past values.

A system is a physical device (or an algorithm) which performs required operation on a discrete time signal.

A discrete time signal is represented as shown in figure below.

Here x(n) input discrete time signal applied to the system. It is also called as excitation.

The system operates on discrete time signal. This is called as procession of input signal, x(n).

Output of the system is denoted by y(n). It is also called as response of the system.

Notation:

When input signal is passed through the system then it is represented by following notation:

Here T is called as transformation operation. Similarly for continuous time system:

Example:

A filter is good example of a system. A signal containing noise is applied to the input of the filter. This is an input signal to the system. The filter cancels or attenuates noise signal. This is the processing of the signal. A noise-free signal obtained at the output of the filter is called as response of the system.

Classification (or properties) of the system:

Properties of systems are with respect to input and output signal. For the simplicity we will assume that the system has single input and single output. But the same explanation is valid for system having multiple inputs and outputs. Important properties (or classification) of the systems are listed below:

In this section we will study some basic operations on continuous time signal:

A) Operation performed on dependent variables:

These operations include sum, product, difference, even, odd, etc.

1) Amplitude scaling:

Amplitude scaling means changing an amplitude of given continuous time signal. We will denote continuous time signal by x(t). If it is multiplied by some constant ‘B’ then resulting signal is,

y(t)= B x(t)

Example:Sketch y(t) = 5u(t)

Solution:we know that u(t) is unit step function. So if we multiply it with 5, its amplitude will become 5 and it shown as follows:

2) Sum and difference of two signals:

Consider two signals x1(t) and x2(t). Then addition of these signals is denoted by y(t)=x1(t)+x2(t). similarly subtraction is given by y(t)=x1(t)-x2(t).

Example:Sketch y(t) = u(t) – u(t – 2)

Solution:First, plot each of the portions of this signal separately

• x1(t) = u(t) …….Simply a step signal

• x2(t) = –u(t-2) ……. Delayed step signal by 2 units and multiplied by -1.

Then, move from one side to the other, and add their instantaneous values:

3) Product of two signals:

If x1(t) and x2(t) are two continuous signals then the product of x1(t) and x2(t) is,

Y(t) = x1(t)x2(t).

Example: Sketch y(t) = u(t)·u(t – 2)

Solution:First, plot each of the portions of this signal separately

• x1(t) = u(t) _ Simply a step signal

• x2(t) = u(t-2) _ Delayed step signal

Then, move from one side to the other, and multiply instantaneous values:

4) Even and odd parts:

Even part of signal x(t) is given by,

And odd part of x(t) is given by,

B) Operations performed on independent variables:

1) Time shifting:

A signal x(t) is said to be ‘shifted in time’ if we replace t by (t-T). thus x(t-T) represents the time shifted version of x(t) and the amount of time shift is ‘T’ sec. if T is positive then the shift is to right (delay) and if T is negative then the shift is to the left (advance).

Example:Sketch y(t) = u(t – 2)

solution:

2) Time scaling:

The compression or expression of a signal in time is known as the time scaling. If x(t is the original signal then x(at) represents its time scaled version. Where a is constant.

If a> 1 then x(at) will be a compressed version of x(t) and if a< 1 then it will be a expanded version of x(t).

In the analysis of communication system, standard test signals play very important role. Such signals are used to check the performance of the system. Applying such signals at the system; the output is checked. Now depending on the input-output characteristic of that particular system; study of different properties of a system can be done. Some standard test signals are as follows:

Delta or unit impulse function

Unit step signal

Unit ramp signal

Exponential signal

Sinusoidal signal

Delta or unit impulse function:

A discrete time unit impulse function is denoted by δ(n). Its amplitude is 1 at n=0 and for all other values of n; its amplitude is zero.

δ(n)= 1 for n=0

δ(n)=0 for n≠0

In the sequence form it can be represented as,

δ(n) = {….,0,0,0,1,0,0,0….} or δ(n) = {1}

The graphical representation of delta function for D.T. signal is as shown in figure below:

A continuous time delta function is denoted by δ(t). mathematically it is expressed as follows:

δ(t)=1 for t=0

δ(t)=0 for t≠0

The graphical representation of delta function for C.T. signal is shown in figure below:

Unit step signal:

Unit step signal means the signal has unit amplitude for positive axis and has zero amplitude for negative axis.

There are two types of unit step signal as follows:

1) Discrete time unit step signal

2) Continuous time unit step signal

Let us discuss these two types one by one:

1) Discrete time unit step signal

A discrete time unit step signal is denoted by u(n). its value is unity (1) for all positive values of n. that means its value is one for n ≥ 0. While for other values of n, its value is zero.

u(n)= 1 for n ≥ 0

u(n)= 0 for n < 0

In the form of sequence it can written as,

u(n) = {1,1,1,1,….}

Graphically it can be represented as follows:

2) Continuous time unit step signal

A continuous type unit step signal is denoted by u(t). mathematically it can be expressed as,

u(t)= 1 for t ≥ 0

u(t) = 0 for t < 0

it is shown in figure below:

Unit ramp signal:

A discrete time unit ramp signal is denoted by ur(n). its value increases linearly with sample number n. mathematically it is defined as,

Ur(n)= n for n ≥ 0

Ur(n) = 0 for n < 0

From above equation, it is clear that the value of signal at a particular interval is equal to the number of interval at that instant. for example; for first interval signal has amplitude 1, for second it has amplitude 2, for third it is 3, and so on.

Graphically it is represented in figure below:

A continuous time ramp type signal is denoted by r(t). mathematically it is expressed as,

r(t) = 1 for t ≥ 0

r(t) = 0 for t < 0

From above equation, it is clear that the value of signal at a particular time is equal to the time at that instant. for example; for one second signal has amplitude 1, for two second it has amplitude 2, for third it is 3, and so on.

It is shown in figure below:

Exponential signal:

In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.

A discrete time exponential signal is expressed as,

Here ‘a’ is some real constant.

If ‘a’ is the complex number then x(n) is written as,

Here Ѳ denotes the phase. Now depending upon value of Ѳ we have different cases:

case 1) : when a>1

a>1

Since the signal is exponentially growing; it is called rising exponential signal.

case 2) : when 0<a<1

Since the signal is exponentially decreases; it is called decaying exponential signal.

case 3) : when a<-1

As shown in figure above, both signals are growing hence it is called as double sided growing exponential signal.

case 4) : when -1<a<0

As shown in figure above, both signals are decreases hence it is called as double sided decaying exponential signal.

If we talk about continuous time exponential signals; everything is same as that of discrete time signals except it has continuous amplitude.

Sinusoidal waveform:

A sinusoidal signal has the same shape as the graph of the sine function used in trigonometry. Sinusoidal signals are produced by rotating electrical machines such as dynamos and power station turbines and electrical energy is transmitted to the consumer in this form. In electronics, sine waves are among the most useful of all signals in testing circuits and analyzing system performance.

There are two types of sinusoidal waveforms:

1) Discrete time sinusoidal waveforms, and

2) Continuous time sinusoidal waveforms.

Let us discuss these waveforms one by one:

Discrete time sinusoidal wave:

Signals typically represent physical variables such as displacement, velocity, pressure, energy, …

In most cases, we are concerned with variables that are time-dependent.

The discrete-time or digital time index is generally specified by t_{n} = n T, where n is an integer and T is the sampling interval or period.

It is common to drop the explicit reference to T (or assume T = 1) and index discrete-time signals by the letter n.

A discrete-time signal which is only dependent on time can be represented by x[n] for n = 0, 1, 2, …

A discrete time sinusoidal waveform is denoted by,

x(n) = A sin (wn)

Here a = amplitude and w = angular frequency = 2πf

This waveform is as shown in figure below:

Similarly a discrete time cosine waveform is expressed as,

x(n) = A cos (wn)

Continuous time sinusoidal signals:

The sinusoidal signals include sine and cosine signals. They are as shown in figure below:

In a communication system, the word ‘signal’ is commonly used. Therefore we must know its exact meaning.

Mathematically, signal is described as a function of one or more independent variables.

Basically it is a physical quantity. It varies with some dependent or independent variables.

So the term signal is defined as “A physical quantity which contains some information and which is function of one or more independent variables.”

Classification of signals:

There are various types of signals. Every signal has its own characteristics. The processing of signals mainly depends on the characteristic of that particular signal. So classification of signal is necessary. Broadly the signals are classified as below:

Continuous and discrete time signals

Continuous valued and discrete valued signals

Periodic and non-periodic signals

Even and odd signals

Energy and power signals

Deterministic and random signals

Multichannel and multidimensional signals

Continuous and discrete time signals:

Continuous signal:

A signal of continuous amplitude is called continuous signal or analog signal. Continuous signal has some value at every instant of time.

Examples:

Sine wave, cosine wave, triangular wave etc. similarly some electrical signals derived from physical quantities like temperature, pressure, sound etc. are also an examples of continuous signals.

Mathematical expression:

Mathematically a continuous signal can be expressed as,

x(t)=A sin(wt+Ѳ)

Here A= amplitude of signal

w = angular frequency=2πf

Ѳ= phase shift

Characteristics:

For every fix value of t, x(t) is periodic in nature.

If the frequency (1/t) is increased then the rate of oscillation also changes.

Discrete time signal:

In this case the value of signal is specified only at specific time. So signal represented at “discrete interval of time” is called as discrete time of signal.

The discrete time signal is generated from continuous time signal by using the sampling operation. This process is shown in figure below.

Consider a continuous analog signal as shown in figure a). This signal is continuous in nature from –infinity to +infinity.

The sampling pulses are shown in figure b). These are train of pulses. Here the samples are taken with Ts as sampling time.

Figure c) shows the discrete time signal

For signal shown in figure a), the expression is x(t)=A cos (wt)

And for signal shown in fig c) , the expression is x(t)= A cos (wn)

Characteristics:

Discrete time sinusoidal signals are identical when their frequencies are separated by integer multiple of 2π.

If the frequency of discrete time sinusoidal is rational number, then such signal is periodic in nature.

For the discrete time sinusoidal, the highest oscillation is obtained when angular frequency w=+π or –π.

Continuous valued or discrete valued signals:

Continuous valued signals:

If the variation in the amplitude of signal is continuous then, it is called continuous valued signal. Such signals may be continuous or discrete in nature. Following figure shows the examples of continuous valued signals.

Discrete valued signals:

If the variation in the amplitude of signal is not continuous but the signal has certain discrete amplitude levels then such signal is called as discrete valued signal. Such signal may be again continuous or discrete in nature as shown in figure below.

Periodic and Non-periodic signals:

Periodic signal:

A signal which repeats itself after a fixed time period or interval is called as periodic signal. The periodicity of continuous time signal can be defined mathematically as,

x(t)=x(t+T0)

This is called as condition of periodicity. Here T0 is called as fundamental period. That means after this period signal repeats itself.

For the discrete time signal, the condition of periodicity is,

x(n)=x(n+N)

Here number ‘N’ is the period of signal. The smallest value of N for which the condition of periodicity exists is called fundamental period.

Following figure shows the examples of periodic signals:

Non-periodic signals:

A signal which does not repeat itself after a fixed time period or does not repeat at all is called as non-periodic or aperiodic signal.

In other words we can say that, the period of non-periodic signal is infinity.

Following figure shows the non-periodic signal:

Even and Odd Signals:

Even signals:

An even signal is also called as symmetrical signal. A continuous time signal x(t) is said to be even or symmetrical if it satisfies the following condition:

Condition for symmetry: x(t)=x(-t)……..for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. That means x(-t) is the signal which is reflected about vertical axis.

Condition for symmetry: x(n)=x(-n)…….for discrete time signal.

Following figure shows the even signal:

Odd signal:

A continuous time signal x(t) is said to be odd signal if it satisfies following condition:

Condition for odd signal: x(-t)=-x(t)…….for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. While –x(t) indicates that the amplitude of the signal is negative. Thus odd signal is not symmetric about vertical axis.

Condition for odd signal: x(-n)=-x(n)……for discrete time signal.

Following figure shows the odd signal:

Note: amplitude of odd signal at origin is always zero.