Filter design technique

A filter can be classified in several different groups, depending on what criteria are used for in classification and also for the design to different techniques as well as algorithms. A filter is a system that passes certain frequency component and totally rejects all others. A filter can be used in many application like simulation, bandwidth limiting, signal processing, image processing based on a special type of application, so the design can be implemented based on which type of application. In this article gives the information about how to design in a filter to know more details about filtering.

Typical filter design requirement :

• The filter should a specific frequency response
• The filter should be causal
• The filter should be stable
• The filter should have a specific impulse response
• The filter should have a specific group delay or phase shift 
• Filter design techniques should be implemented in particular software or hardware

Filter design techniques depend on this specification :

• The approximation of the specification using a causal discrete time system or design
• The realization of the system
• And one the specification of the desired properties of the system

Digital filter algorithms

In both recursive and non-recursive (FIR full form and IIR full form filter) in digital signal possessing, the algorithm can be implemented is based on DFT meaning or differential equation.

The algorithm would be used in filtering application in two specific areas like: in filtering algorithms and in signal analysis algorithms.

Their implementation is possible by using hardware or some software. Filtering algorithm can be implemented software like Matlab.

The figure below represents the diagram of the cascaded filter structure. It will help to better to understand how the comparability condition to be work.

Cascaded filter design

IIR filter applications

IIR filter stands for infinite impulse response are also known as the recursive type of filter operates on the current and past input value and current and past output value. In theoretically the impulse response of an IIR filter never reaches zero and is an infinite response.

• Telecommunication 
• Clock recovery in data communication
• Receiver anti-imaging filter
• Digital telephony called digital dual tone multi-frequency touch-tone receiver
• Signal monitoring application

Digital filter basics

Let us know about digital filter first let us learn about filtering. Filtering is the special part of the digital signal processing system. It can also remove unwanted part of the signal such as random noise. 

Digital filter uses a digital filter processor, it is a specific characteristic that you need to pay special attention to. In general digital filter can be considered two types are known as IIR and FIR filter. In the both of recursive and non-recursive (FIR and IIR filter) in DSP, the algorithm can be implemented is based on DFT or differential equation.

In general analogue input signal must satisfy certain requirements, That will be converting an output digital signal into analogue signal form.

A system that performs the mathematical operation in signal processing on a sampled signal to reduce or enhance certain aspects of that signal is known as a digital filter. It is removed the unwanted parts of the signal.

Explore more information:

1. IIR filter basics 
2. FIR filter basics 
3. Digital filter types 

Digital filter types

Digital filter meaning can be classified in several different groups but there are two major types of digital filter are to be full form of FIR  and full form of IIR. FIR is also called recursive, IIR is also called non-recursive types of filter.

In this article, we have to discuss the basic two types of filter known as the FIR and IIR filter. Now let us check it out some basic characteristics of FIR and IIR filters.

Characteristics of FIR filter :

• Linear phase characteristics
• Stability
• High filter order

Characteristics of  IIR filters :

• Non-linear phase characteristics
• Low filter order
• This filtering, the resulting digital filter has the potential to become unstable 

There is also some  other filter like :

• High pass
• Bandpass
• Low pass
• Stopband
• Notch
• Comb filter 
• All pass filter

Explore more information:

1. Filter design techniques 
2. Advantages and disadvantages of digital filter 
3. Digital filter algorithm 
4. Digital filter applications 
5. Block diagram of FIR filter 
6. FIR filter characteristics 
7. Butterworth low pass filters design 
8. Advantages and disadvantages of FIR filters 
9. IIR filter properties 
10. IIR filter application 
11. Advantages and disadvantages of the bi-linear transformation method 
12. IIR filter examples 
13. Impulse invariant method example 
14. Advantages and disadvantages of IIR filter 

IIR filter basics

IIR stands for infinite impulse response, IIR filter is one of the two primary types of digital filter. It is a very important part of a digital signal processing application. The impulse response of the IIR filter is to be infinite because of feedback in the filter; If you put in an impulse an infinite number of non zero values will come out.

IIR filter is that it can achieve a given filtering characteristic using fewer calculations and memory than a similar FIR filter.

One of the disadvantages of IIR filter is harder to implement using fix point arithmetic also don't offer the computational advantages of FIR filters for multi-rate applications.

IIR full form

What is the full form of IIR?

• Infinite Impulse Response 

What does IIR mean?

IIR filter is the most efficient and fundamental element of a filter to implement in digital signal processing. It is properly applying to many linear invariant systems. IIR filter can achieve a given filtering characteristic using less calculation and memory than a similar for FIR filter.

Explore more information:

1. Full form of DSP
2. Full form of DTFT
3. Full form of LTI
4. Full form of FIR
5. Full form of IIR
6. Full form of ROC
7. Full form of DFT

FIR filter characteristics

Most of the FIR full form filter is that linear phase filter When the linear phase filter is desired an FIR filter meaning is usually used. Here this article gives information about some characteristics of FIR filter to know more about details in FIR filters.

• Can be adaptive
• Unconditionally stable
• No analog equivalent
• In this filter, there is no dispersion hence no change in overall signal shape due to non-linear phase shift
• Impulse response has a finite duration
• Linear phase, constant group delay
• Easy to understand and design like widows sinc method, Fourier series expansion with windowing, frequency sampling using inverse FFT 

FIR full form in dsp

What is the full form of DSP?

• Finite Impulse Response

What does DSP mean?

FIR filter whose response to any finite length input is of finite duration. It can easily be designed to be linear phase. One of the most DSP microprocessors, the FIR calculation can be easily done by looping a single instruction. FIR filter can be discrete time or continuous time as well as digital or analog.

Explore more information:

1. IIR full form

Digital filter applications

In digital filtering, The function of a filter is removing the unwanted part of the signal, such as random noise or extract useful part of the signal such as a component lying within a certain frequency range. This article gives information about the application of a digital filter to know more advance application in Filtering in details.

1. Simulation/modeling 

• Simulating communication channels
• Modeling the human auditory system

2. Bandwidth limiting

• Anti-aliasing filters for sampling
• Ensuring that a transmitted signal occupies only its allotted frequency band

3. Noise suppression

• Imaging devices
• Bio-signals like heart, the brain response

4.  Image processing

• Image processing application
• Enhancement of selected frequency ranges

5. Signal processing

• Speech synthesis
• Geophysical signal
• Processing of seismic
• Equalizers for audio signals
• The audio system such as CD/DVD players
• Removing the DC component of a signal

6. Special operation

• Differentiation
• Integration
• Hibert transform

What is frequency warping

Frequency warping is a one types of transformation process where one spectral representation on a certain frequency scale measured in Hz.  

The amplitude response of digital IIR filter is expanded at lower frequencies and compressed at higher frequencies in comparison to the analog filter. This effect is called frequency warping.

full form of DTFT

What is the full form of DTFT?

• Discrete-Time Fourier Transform 

What does DTFT mean?

DTFT is one type of Fourier transform analysis, it is only applicable to the uniformly spaced of continuous device function. The same way the inverse IDTFT is the original sampled data sequence, the IDFT is a periodic summation of the original sequence.

Full form of ROC

What is the full form of ROC?

• Region Of Convergence 

What does ROC mean?

It is one of the most important roles in the use of z transform for analysis of signal and system. All complex values for which the integral in the definition converges form a region of convergence (ROC) in the s-plane. ROC does not contain any poles.

What is LTI system

LTI is also called an LSI system. LTI stands for the linear time-invariant system also stands for a linear shift-invariant system. LTI is a class of system used in signal and system in digital communication for a linear and time-invariant system.

Linear system is the system whose outputs for a linear combination of input are the same as a linear combination of input is as a linear combination of individual response to those inputs and also have the time-invariant system are a system where the output does not depend on when the input was applied.

Linear convolutions in DSP

Consider relaxed LTI system. A relaxed system means if input x (n) is zero then output y (n) = 0 is zero. Let us say unit impulse 𝛿 (n) is applied to this system then its output is denoted by h (n). h (n) we called as the impulse response of the system.

Step 1 :

         T
𝛿 (n) → y (n) = h (n)

Step 2 :

             T
𝛿 (n-k)  → y (n) = h (n-k)

Step 3 :

                       T

x (k) 𝛿 (n-k)    →  y (n) = x (k) h(n-k) 

Step 4 :

 ∞                                       T   ∞  

∑        = x (k) 𝛿 (n-k) =   y (n) →  ∑      x (k) h(n-k)

k= -∞                                             k= -∞

y (n) = 

 ∞                                        

∑        = x (k) h (n-k)  

k= -∞           

                   
                          ∞

x (n) * h (n)  =   ∑      x (k) h(n-k)

                          k= -∞                                            

                          

Properties of linear convolution

Linear convolution or proof of LTI system is completely characterized by unit impulse response h(n).

These properties are :

• Commutative property
• Associative property
• Distributive property

1.  Commutative property :

x (n) * h (n) = h (n) * x (n)

2. Distributive property :

x (n) * [ h₁ (n) +  h₂ (n) ] =  [ x (n) * h₁ (n) ]  + [   x (n) * h₂ (n) ]

3. Associative property :

[ x (n) * h₁ (n) ] *  h₂ (n)  =  x (n) * [ h₁ (n) *  h₂ (n) ] 

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 :