Basic signal operations

Time shifting operations :

The different time-shifting operations are as follow :

• Time delay
• Folding
• Time advance
• Folding
• Folding and advance
• Folding and delay

1. Time delay :

In the case of a discrete time signal, the given sequence can be delayed by a few samples. We know that the discrete time signal is denoted by x (n). 

Suppose we want to delay this sequence by "k" sample. It will be denoted by x (n-k).

x (n) = Original sequence 

x (n-k) = Original sequence delayed by k samples.

Here k is an integer

x(n) = {1, 2, 3, 4, 5 },  k=2

            ↑

x(n-2) = { 0, 0,1, 2, 3, 4, 5 }

                ↑

2. Time advance : 

Time advance operation is opposite to the time delay operation. Consider the same sequence is shown  given below :

x(n) = {1, 2, 3, 4, 5 }

            ↑

x(n+2) = { 1, 2, 3, 4, 5 }

                         ↑

3. Folding :

Folding is also called as reflection. Thus if x (n) represents input signal then x (-n) represent folded input signal.

x (n) = { 1, 2, 3, 4, 5 }

x  (-n) = { 5, 4, 3, 2, 1 }
                               ↑

4. Folding and delay :

• First fold the sequence x(n); that means obtain x (-n)
• Then delay the folded sequence by k sample

         delay
x (n)   →    x (n-k)
          
           delay
x (-n)    →    x [- (n-k) ] = x (-n+k)

x (n) = { 1, 2, 3, 4, 5 }
               ↑

x  (-n) = { 5, 4, 3, 2, 1 }
                                ↑

x (-n+2) =  { 5, 4, 3, 2, 1 }
                           ↑

5. Folding and advance  :

         advance
x (n)     →        x (n+k)

        advance
x (-n)     →        x [- (n+k) ] = x (-n-k)

x (n) = { 1, 2, 3, 4, 5 }
               ↑

x  (-n) = { 5, 4, 3, 2, 1 }
                                ↑

x(-n-2) = {  5, 4, 3, 2, 1, 0, 0 }

                                          ↑

          
Time scaling operation :

• Downscaling
• upscaling

1. Down scaling :

consider the same sequence x (n)  = { 1, 2, 3, 4, 5 }
                                                              ↑

y (n) = x (2n)

Now from given sequence x (n)  we can write :
x(0) = 1
x(1) = 2
x(2) = 3
x(3) = 4
x(4) = 5

This gives the value of x (n) for different value of n;

y(n) = x (2n)
y(0) = x (0) = 1
y(1) = x (2) = 3
y(2) = x (4) = 5
y(3) = x (6) = 0

y(n) = x (2n) = {1, 3, 5, 0.....}
                         ↑

2. Up scaling or expansion :

Consider same input sequence x (n) =  { 1, 2, 3, 4, 5 } is applied to certain device which produces output y(n) = x(n/2).             ↑

Thus in this case :

y(n) = x(n/2) 

y(0) = x(0/2) = x(0) =1
y(1) = x(1/2) → No sample
y(2) = x(2/2) = x (1) = 2
y(3) = x(3/2)= x(1.5) → No sample
y(4) = x(4/2) = x (2) = 3
y(5) = x(5/2) = x (2.5)  → No sample
y(6) = x(6/2) = x (3) = 4
y(7) = x(7/2) = x (3.5) → No sample
y(8) = x(8/2) = x (4) = 5

y (n) = x (n/2) = { 1, 0, 2, 0, 3, 0, 4, 0, 5 }
                            ↑

Amplitude scaling operation :

• Up-scaling 
• Down-scaling
• Addition
• Multiplication

1. Up-scaling :

x (n) = { 1, 2, 3, 4, 5 }
               ↑
x(0) = 1
x(1) = 2
x(2) = 3
x(3) = 4
x(4) = 5

y (n) 2 x(n) = { 2, 4, 6, 8, 10 }
                        ↑

2. Down-scaling :

x (n) = { 1, 2, 3, 4, 5 }
              ↑

y (n) = x (n) / 2 = { 0.5, 1, 1.5, 2, 2.5 }
                                ↑

3. Addition :

x₁(n) = { 1, 1, 0, 1, 1 }

                     ↑

x₂(n) = { 2, 2, 0, 2, 2 }

                     ↑

y (n) = x₁(n) + x₂(n) 

y (n) = { 3, 3, 0, 3, 3 }
                     ↑

4. Multiplication :

x₁(n) = { 1, 1, 0, 1, 1 }

                     ↑

x₂(n) = { 2, 2, 0, 2, 2 }

                     ↑

y (n) = x₁(n) * x₂(n) 

y (n) = { 2, 2, 0, 2, 2 }
                     ↑                                              

Power of energy signal

Let x (t) be an energy signal. x(t) has a finite non zero energy. Let us calculate the power of x(t). By definition, stated equation the power of x(t) is given by :

Power P = 

P = 0 * E = 0 

What is system

A system is defined as the entity that operate one or more signals to accomplish a function, to produce new signals.

It is collection of highly integrated  parts designed to achieve a desired outcome. The system has wide variety of input which are manipulated through a process or series of process in a manner to produce an output.

Types of system :

1. Communication system
2. Biomedical signal processing
3. Auditory system
4. Control system
5. Remote sensing system

Symmetrical and Anti symmetrical signal

Symmetrical signal or even signal :

A signal x (t) is said to be symmetrical or even if it satisfies the following condition.

Condition for symmetry : x(t) = x(-t)

Where,  x(t) = Value of the signal for positive "t" and x(-t) = Value of the signal for negative "t".

An example of symmetrical signal is a cosine wave shown in figure : This article also gives the some properties of even and odd signals 

Anti symmetrical signal or odd signal :

A signal x(t) is said to be Anti symmetrical or odd if it satisfies the following condition,

Condition for anti-symmetry : x (t) = -x(-t)

An example of odd signal is a sine wave shown in figure :

Advantages and disadvantages of FIR filters

FIR short terms is finite impulse response. FIR filter are called as non-recursive filters because they do not use the feedback.This article gives the information about advantages and disadvantages of  FIR filters to know more details about it. 

Advantages of  FIR filters :

• FIR filter are always stable
• It is simple
• FIR filter is having linear phase response
• It is easy to optimize
• Noncausal
• Round of noise error is minimum
• Both recursive, as well as nonrecursive filter, can be designed using FIR designing techniques
• For designing a filter having any arbitrary magnitude response; FIR designing techniques can be easily applied
• Good performance
• Robust
• The necessity of computational techniques for filter implementation
• The requirement of large storage
• The incapability of linear phase response

Disadvantages of FIR filters :

• Large storage requirements
• Can not simulate prototype analog filter
• For the implementation of FIR filter complex computational techniques are required
• It is hard to implementation than IIR
• Expensive due to large order
• Require more memory
• Time-consuming process

Explore more information:

1. Advantages and disadvantages of digital filter
2. Advantages and disadvantages of IIR filter
3. Advantages and disadvantages of active filter
4. Advantages and disadvantages of passive filter
5. Advantages and disadvantages of FIR and IIR filter

Energy of power signal

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

Energy E = 

               

 

              E = ∞

Advantages and disadvantages of bilinear transformation method

Bi-linear transformation method is one of the most important method is that transforming the analog filter into appropriate IIR filter. So here this article gives the information about advantages and disadvantages of bi-linear transformation to know more details about it.

Advantages of bi-linear transformation method  :

• The mapping is one to one
• There is no aliasing effect
• Stable analog filter is transformed into the stable digital filter
• There is no restriction one type of filter that can be transformed
• There is one to one transformation from the s-domain to the Z- domain

Disadvantages of bi-linear transformation method :

• The mapping is non linear in this method because of this frequency warping effect takes place

Full form related to Digital signal processing

DSP - Digital signal processing

ROC - Region of convergence

DT - Discrete time

FT - Fourier transform

DFT - Discrete Fourier transform

DTFT - Discrete time Fourier transform

IDFT - Inverse  discrete Fourier transform

FFT - Fast Fourier transform

BLT - Bi-linear transformation

LTI - Linear time invariant 

LCCDE - Linear constant coefficient difference equations
DIT - Decimation in time
TVP - Time variance property

PFE - Partial function expansion 

IIR - Infinite impulse response

FIR - Finite impulse response

LTI full form

What is the full form of LTI?

• Linear Time-Invariant 

LTI theory comes from applied mathematics and has some direct application like in signal processing, control theory, NMR spectroscopy seismology, and some other technical areas. It is a system that obeys the linear property and time-invariant property.

Example :

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

y(n,k) = x (n-k) -x (n-k-1).............(1)

Replace n by n-k throughout the given equation above :

y (n-k) = x (n-k) - x(n-k-1)............(2)

Compare equations (1) and (2)

Here y (n,k) = y (n-k).

Thus the system is time-invariant.

Explore more information:

1. Full form of DSP

Properties of LTI systems

Properties of system are with respect to input and output signal. Important properties of systems are as follows : 

• Dynamic or static system
• Time variance
• Linearity
• Causality
• Stability

1. Static or dynamic system :

(I) Static system :

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

Example :

y (n) : 7 x (n)

here 7 is constant which multiplies input x (n) but output at nth instant that means y (n) depends on input at the same (nth) time constant x (n). so this is a static system.

(II) Dynamic system :

It this system output at any instant of time depends on input sample at the same time as well as at other times.

Example :

Note that if x (n) represents input signal at present instant then,

past and future signal :

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

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

y (n) : x (n) + 6 x (n-1)

here x (n-1) is previous  sample. so system is dynamic.

y (n) = 5 x (n+2) + x (n)

here (n+2) indicates advanced version of input sample that means it is future sample; so tis is dynamic system.

2. Time variant or time invarient systems :

A system is time invarient if its input output characteristics do not chance with time.

A time invarient system means its input output characteristics are not changing with time shifting.

Let us consider we are applying input signal x (n) to the system and produces output y (n).

Now delay input by k samples. that means input becomes x (n-k). Apply this delayed input to the same system . Let us say the system now gives output y (n-k). then this system is called as time or shift invarient system.

Observe that initially the output is y (n) with input x (n). When input is delayed by "k" samples then the output is also delayed by the same "k" samples. Thus the input-output characteristics are not changed . so the system is invarient.

Example :

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

y(n,k) = x (n-k) -x (n-k-1).............(1)

Replace n by n-k throughout the given equation :

y (n-k) = x (n-k) - x(n-k-1)............(2)

Compare equations (1) and (2)

Here y (n,k) = y (n-k).

Thus the system is time invarient.

3. Causal or anticausal system :

(I) Causal system :

A system is said to be a causal if output at any instant of time depends only an present and past inputs. But the output does not depend on future inputs.

Example :

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

y (n)  = 6 x(n)

(II) Anticausal systems :

A system is said to be anticausal if its output depends not only on present and past inputs but also on future inputs.

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

y(n) =  x(n) + n x(n+1)

4. Stable or Unstable system :

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 of value. So bounded input means input signal is having some finite value.

(I) Stable system :

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

Here a relaxed system means when input to the system is zero then the output of system is also zero.

Example :

T[ x (n) ]  = a x(n) +5

(II) Unstable system :

An initially relaxed system is said to be unstable if bounded input produces unbounded output.

When unstable system is practically implemented then it causes overflow.

Unstable system shows erratic and extreme behavior.

Example : 

5. Linear and non linear system :

When input x (n) is zero then output y (n) = 0. Thus the system is linear. Now first step is satisfied so we will check remaining step for linearity.

Example :

T x(n) = a x(n) + 6 

step 1 : put x (n) = 0 

y (n) = 0 + 6

step 2 : 

x₁ (n) = y₁ (n) = a x₁(n) + 6

x₂(n) = y₂(n) = a x₂(n) + 6

y ‘(n)  = a₁ y₁ (n) + a₂ y₂ (n)

y '(n)  = a₁ [a x₁(n) + 6 ]  +  a₂ y₂ (n) + a₂ [ a x₂(n) + 6 ]........................(1)

step 3 :

a₁  x₁(n) + a₂  x₂(n)  = a [ a₁  x₁(n) + a₂  x₂(n) ] + 6 

y'' (n) = a [ a₁  x₁(n) + a₂  x₂(n) ] + 6......................................................(2)

step 4 : compare  equation (1) and (2)

y '(n) = y'' (n) ; the system is non linear.

Difference between energy and power signals

An energy signal has zero average power, whereas a power signal has infinite energy. Here this article gives the information about difference between energy and power signal to know more details about it.

• The signal having finite non zero power are called as power signal while the signal having a finite non zero energy are called as energy signal.
• On the other hand a signal is refered as a energy signal, if and only if energy of the signal satisfies the condition 0 < E < ∞ and same way also referred to as a power signal, if and only if the average power of the signal satisfies the condition 0 < P < ∞. 
• Example of power signal is sinusoidal, unit step etc while in energy signal are to be exponentially decaying or increasing signal.
• Almost all the periodic signals in practice are power signals and almost all the non periodic signals are energy signals.
• Power signals can exist over an infinite time. They are not time limited while  in energy signal exist over a short period of time. They are time limited.
• Energy of a power signal is infinite but in power of an energy signal is zero.

Periodic and non periodic signals

Periodic signals :

• A CT signal which repeats it self after a fixed time period is called as a periodic signal. The periodicity of a CT signal can be defined mathematically as follows :

x(t) = x(t+T₀)

Where ; 

T₀ = periodic of signal x(t) 

periodic signal are sine wave, cosine wave, square wave etc.

Periodic signal

Non periodic signals :

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

The non periodic signal do not satisfy the condition of periodicity stated in equation :

x(t) ≠  x(t+T₀)

Sometimes it is said that an aperiodic signal has a periodic T₀  = ∞. Figure shows a decaying exponential signal.

This exponential signal is non periodic but it is deteministic because we can mathematically express it as x(t) = e^-ɖt.

Non periodic signal

Digital signal processing based project

• An automatic speaker recognition system
• Area time efficient scaling free CORDIC using generalized micro rotation selection
• Investigation in FIR filter to improve power efficiency and delay reduction
• Design and implementation of adaptive filtering algorithm for noise cancellation in speech signal on FPGA
• Fault-tolerant parallel filters based on error correction codes
• Practical energy aware link adaption for MIMO-OFDM system
• Maximize network topology lifetime using mobile node rotation
• A reconfigurable smart sensor interface for industrial WSN in IoT environment
• High throughput programmable systolic array FFT architecture and FPGA implementations
• A reconfigurable overlapping FFT/IFFT filter for ECG signal de-noising
• Subband adaptive filter for acoustic echo cancellation
• FPGA based partial reconfigurable FIR filter design
• Energy efficient spectrum access in cognitive radios
• Radio interface evolution towards 5G and enhance local area communications
• Power optimization of single precision floating point FET design using fully combinational circuits
• Adaptive variable step size in LMS algorithm using evolutionary programming
• Functional link adaptive filter for non linear acoustic echo cancellation
• Design and FPGA implementation of linear FIR low pass filter based on Kaiser window
• Low latency systolic Montgomery multiplier for finite field based on pentanomials
• A review on FPGA based pulse processing system
• An FPGA implementation of frequency output
• Analysis and implementation of low cost FPGA based digital pulse width modulators
• Achieving energy efficiency and reliability for data dissemination in duty cycle WSNs 
• Joint virtual MIMO and data gathering for wireless sensor network

Basic signals

In signals and system, we need to use some standard or elementary signals. In this section, we will show some important standard signal graphically and express them mathematically.

Some of the standard continuous and discrete time signals are :

• A DC signal
• Unit step signal
• Delta or unit impulse function
• Sinusoidal signal
• Exponential function
• Signum function
• Sinc function

A DC signal :

A DC signal is shown in the figure. As seen from the figure or waveform the amplitude A of a direct current signal remains constant independent of time.

A DC signal is x (t) = A  - ∞ < t <  ∞

Sinusoidal signal :

The sinusoidal signal includes sine and cosine signals.

Mathematically the can be represented as follows :

A sine signal   x(t) = A sin ωt = A sin (2∏ft )
Same way in a cosine signal   x(t) = A cos ωt = A sin (2∏ft )

Here = A = Amplitude 
ω = Angular frequency = 2∏f

Unit step signal :

The unit step signal is as shown in the figure. It has a constant amplitude of unity(1) for the zero of the positive value of time "t" . Whereas it has zero value for a negative value of t.

The unit step signal is mathematically represented as, 

Unit step signal called as    u (t) = 1  for t > 0

                                                   = 0  for t < 0

Signum function :

The signum function is as shown in figure. It is represented mathematically as follows :

sgn(t) =  1 for t > 0 
          = - 1  for t < 0 

Delta or unit impulse function :

The delta function is an extremely function used for the analysis of the communication system. The impulse response of a system is its response to a delta function applied at the input signal.

Delat function : = 0 for t ≠ 0 

 Unit ramp function :

A continuous time unit ramp signal is denoted by ramp called r (t). Mathematically it is expressed as, 

r (t) = t for t >= 0
         = 0 for t< 0

What is signal

A signal is basically an electrical or electromagnetic current that is used for carrying data from one device or network to another. It is a physical quantity. It varies with some dependent or independent variables. 

So term of Signal can be defines as "A physical quantity which contain some information and which is function of one or more independent variables."

A signal can be analog type or digital type. Signal basically one dimensional and two dimensional. In one dimensional signal the function depends on a signal variable, i.e. speech signal whose amplitude varies with time while in multi dimensional signal depends on two or more variables, i.e an image because it is horizontal and vertical co-ordinates. 

DSP application

In DSP stands for digital signal processing, the Signal definition can be defined as "A physical quantity which contain some more information and which is a function of one or more independent variables."Digital signal processing" takes real-world signal like audio, video, pressure, voice that have been digitized and then mathematically manipulate them. Mathematical processing based algorithm can be easily implemented using DSP. 

Space :

• Space photograph enhancement
• Data compression
• Intelligent sensor analysis

Medical :

• Electrocardiogram analysis
• Medical image storage and retrieval
• Diagnostic imagine (MRI,CT, Ultra sound etc)

Military application :

• Radar
• Sonar
• Secure communication

Telephone :

• Signal multiplexing
• Filtering
• Echo reduction
• Video and data compression

Image and video :

• Image and sound compression for Multimedia presentation
• Movie special effect
• Video conferencing calling
• Noise suppression
• Medical imaging
• Bio-metric applications

Industrial :

• CAD and design tools
• Power line monitors
• Robotics
• Spectral analysis 

Communication and Audio :

• Removal speech
• Music applications
• Envelope for audio files

LASER full form

What is the full form of LASER?

Answer :

• Light Amplification By Stimulated Of Radiation 

What does LASER mean?

A LASER is one type of device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. It emits an intense beam of coherent monochromatic light through a process of optical amplification based on the stimulated emission of photons.

It is an electronic device that produces light, actually electromagnetic radiation. This electromagnetic radiation is done through optical amplification.

LASER radiation has very special properties that make it used in different types of applications in daily life. It is used to manufacture a wide variety of electronic devices like CD ROMs, DVD players, barcode readers, etc.

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Application of electric drives

In many of the industrial application, an electric motor is one of the most important components. An electric motor is a most important part, it is an energy transmitting device and the working machine. Here this article gives the application of electric drives to know more details about it. 

• A ceiling fan motor with regulator.
• A motor and also have conveyor belt with material on its belt.
• Food mixer without food is processed.
• It is also used a large number of industrial as well as domestic application.
• It is also used rolling mills, textile mills as well as some machine tools.
• Some other application like pumps, robots, washing etc.
• It can also use various traction like an electric train, electric buses, trams, trolleys, battery driven solar power vehicles.
• Electric propulsion.
• Cement kilns.
• Elevators, escalators, and lift.
• Automotive applications.
• Spindles and servos.
• Plant automation.