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.


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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.


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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 :

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

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.

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