Multichannel and Multidimensional Signals with easy Explaination

Hello friends, in this quick article we are going to learn about Multichannel and Multidimensional Signals with there definitions and day-to-day example.

Multichannel and Multidimensional Signals

Multichannel and Multidimensional Signals
Multichannel and Multidimensional Signals

Multichannel Signals

  • As the name indicates, multichannel signals are generated by multiple sources or multiple sensors.
  • The resultant signal is the vector sum of signals from all channels

Example: A common example of a multichannel signal is ECG waveform. To generate ECG waveform; different leads are connected to the body of a patient. Each lead is acting as an individual channel. Since there are n number of leads; the final ECG waveform is a result of the multichannel signal. mathematically final wave is expressed as,

Multidimensional Signals

If a signal is a function of a single independent variable, the signal is called a one-dimensional signal. On the other hand, if the signal is a function of multiple independent variables then it is called as a multidimensional signal.

A good example of a multidimensional signal is the picture displayed on the TV screen. To locate a pixel on the screen, two coordinates namely x and y are required. Similarly, this point is a function of time also. So to display a pixel, minimum three dimensions are required; namely x, y, and t. Thus this is a multidimensional signal. Mathematically it can be written as P(x, y, t).

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Deterministic and Random Signals

Hi friends, in this article we will learn some basic concepts about Deterministic and Random Signals

Deterministic signal

A signal which can be described by a mathematical expression, lookup table or some well-defined rule is called as the deterministic signal.

Examples: Sine wave, cosine wave, square wave, etc.

Deterministic signal
Deterministic signal

Sine wave can be represented mathematically as,

x(t) = A sin (2πft)

where A is the amplitude of a signal

f = frequency of a signal

Note: The deterministic signals such as sine wave, cosine wave, etc are periodic. But some deterministic signals may not be periodic. The exponential signal is an example of a non-periodic signal.

Random Signal

A signal which cannot be described by any mathematical expression is called as a random signal. Due to this, it is not possible to predict about the amplitude of such signals at a given instant of time.

Example: A good example of a random signal is noise in the communication signal. Below figure shows one of the random signals.

Energy and Power Signal

Hi friends, in this tutorial we are going to learn about Energy and Power signal in Signals and Systems.

Energy and Power Signal

A) Power Signals

There are three power signals:

  1. Instantaneous power
  2. normalized power
  3. Average normalized power

Let’s see each type one by one.

Instantaneous power

An instantaneous power across resistor R is given by

Here V is the voltage across the resistor R. Let us consider that this voltage is represented by x(t) then above equation becomes

Similarly, in terms of current, we can write,

If we say current is denoted by x(t), then we can get

Normalized power

Every time we don’t know if x(t) is a current signal or voltage signal. therefore to make power equation independent of nature of x(t), we will normalize it by putting R = 1, thus the equation 2) and 4) becomes

Average normalized power

average normalized power is given by,

  • Here magnitude of x(t) is written so this equation is also applicable if x(t) is complex
  • Integration is taken from -T0/2 to T0/2 that is it means, it is from entire time period T0
  • Considering T0 as average value, integration term is multiplied by 1/T0

Based on all the above equations we can define power signal as below

Definition of Power Signal

A signal x(t) is said to be power signal is its normalized average power is non-zero and finite. thus for the power signals the normalized average power, P is finite and non-zero.

For the discrete time signals, average power P is given by,

In above equation, it is expected that N>>1

B) Energy Signal

The total normalized energy for a real signal x(t) is given by

but if the signal x(t) is complex then we can write above equation as

Definition of energy signal

A signal x(t) is said to be an energy signal if its normalized energy is non-zero and finite. Hence for the energy signals, the total normalized energy (E) is non-zero and finite.

The energy of discrete time signal is denoted by E, it is given by,

You can remember the conditions for Energy and Power signal using below table.

Power of the energy signals

Let us consider, x(t) is an energy signal. i.e. x(t) has a finite non-zero energy. let us calculate the power of x(t). By definition, explained in equation no. 6) the power of x(t) is given by

Therefore we can say that the power of energy signal is zero over a finite time.

Energy of a power signal

Let us consider, x(t) be a power signal. The normalized energy of this signal is given by,

This equation can be written as,

Therefore we can say that energy of a power signal is infinite over a finite time.

Comparison of Energy and Power Signals

Power Signals Energy Signals
1. The signal having finite non-zero power are called as Power Signals 1. The signals having finite non-zero energy are called as energy signals
2. Almost all the periodic signals in practice are power signals 2. Almost all the nonperiodic signals are energy signals
3. ower signals can exist over an infinite time. They are not time-limited 3. Energy signals exist over a short period of time. They are time limited
4. The energy of a power signal is infinite 4. Power of an energy signal is zero

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Stable or Unstable System (Stability Property)

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 < 8, so it’s a finite value. Then if input is bounded, we can write,

|x(n)| = Mx < 8

Similarly for C.T. system

|x(t)| = Mx < 8

Similarly consider some finite number My whose value is less than infinity. That means My < 8, so it’s a finite value. Then if output is bounded, we can write,

|y(n)| = My < 8

Similarly for continuous time system

|y(t)| = My < 8

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.

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Causal and Noncausal System (Causality Property)

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)

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Linear or Non-linear Systems (Linearity Property)

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.

That means,

T[a1 x1(n) + a2 x2(n)] = a1 T[x1(n)] + a2 T[x2(n)]…………….1)

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:

f[a1 x1(t) + a2 x2(t)] = a1 y1(t) + a2 y2(t)…………….1)

Where a1 and a2 are constants.

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 1st 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:

1

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 2

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.

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Time Variant or Time Invariant Systems

Time Variant or Time Invariant Systems

Definition:

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

1

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

2

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.

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Static or Dynamic Systems

Static or dynamic systems

In the last article we have discussed about introduction to the systems and its properties or classification. Let us study these Properties one by one.

a)      Static systems:

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) = x2(n) + 8x(n) + 17

Here also output at nth 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 nth 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 nth instant depends on input at nth 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.

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Properties of a system

Properties of a system

Introduction:

System:

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.

Discrete time system
Discrete time system
  • 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:

1

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

2

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:

Now based on the properties, the systems can be classified as follows:

Properties of system
Classification of system

We will see these different types (i.e. properties) of a system in upcoming posts. So stay connected.

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Basic operations on continuous time signal

Basic operations on continuous time signal

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:

amplitude scaling
amplitude scaling

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:

Sum and difference of two signals
Sum and difference of two signals

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:

Multiplication of two signals
Multiplication of two signals

4)      Even and odd parts:

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

even

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

odd

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: time shifting

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).

Example:  Let x(t) = u(t) – u(t – 2). Sketch y(t) = x(t/2)

solution:

time scaling 1time scaling 2

time scaling 3

3)      Time reversal (Time inversion):

Flips the signal about the y axis. y(t) = x(-t) .

Example: Let x(t) = u(t), and perform time reversal.

solution: time reversal

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