Impulse invariant method example

Step 1 : Analog frequency transfer function H(s) will be given. If it not given then obtain expression of H(s) from the given specification

Step 2 : If required H(s) by using fraction expansion

Step 3 : Obtain Z transform of each PFE term using in-variance transformation equation

Step 4 : Obtain H(z) this is required digital IIR filter

Find out H(Z) using impulse in-variance method at 5 Hz sampling frequency from H(s) as given below :

H(s) = 2 / (s+1) (s+2)

Step 1 : Given analog transfer function is, 

 H(s) = 2 / (s+1) (s+2)

Step 2 : We will expand H(s) using partial fraction expansion as :

H(s) = A₁/s+1 + A₂/s+2

p₁= -1 and p₂ =-2

Step 3 : 

A₁ = s+1 * 2 / (s+1) (s+2)  where s=-1

A₁ = 2/-1+2

A₁ = 2

Same way 

A₂= s+2 * 2 / (s+1) (s+2)  where s=-2

A₂= 2/-2+1

A₂= -2

H(s) = 2/s+1 -  2/s+2

Step 3 : Obtain Z transform of each PFE term using in-variance transformation equation

1 / s-p_k = 1/ 1-e ^p_k^T_s. Z^-1

^1/ F_s = 1/5 = 0.5 sec = ^T_s

_{1/ s+1  →} 1/ 1-e ^-1(0.2) . Z^-1

_{1/ s+1  →} 1/ 1-e ^-0.2 . Z^-1

_{1/ s+2 →} 1/ 1-e ^-2(.2) . Z^-1

_{1/ s+2 →} 1/ 1-e ^-0.4 . Z^-1

Step 4 : Obtain H(z) this is required digital IIR filter

H(Z) = A₁/ 1-e ^p₁^T_s. Z^{-1    +}  A₂/ 1-e ^p₂^T_s. Z^-1

^{H(Z) =} 2/ 1-e ^-0.2 . Z^{-1    -}  2/1-e ^-0.4 . Z^-1

^{H(Z) =  2/ 1-0.818}Z^-1 - 2/ 1-0.67Z^-1

^{H(Z) = 2Z / Z-0.818 - 2Z/ Z-0.67}

H(Z) = 2Z(Z-.67-2Z (Z-0.818) / (Z-0.818)(Z-0.67)

^{H(Z) = 0.29 Z /} Z²-1.488Z+0.54

This require transfer function for digital IIR filter.

Butterworth low pass filters

There are so many digital filters like FIR and IIR filter, here this article gives one more analog type batter-worth filter. To shown in figure typical characteristics of batter-worth low pass filter.

This type of response is called a butter-worth response because its main characteristics are that the pass-band maximally flat. It means there are no variations in the pass-band device.

Now the magnitude squared response of low pass butter-worth filters is given by,

  │H(Ω)²│  =   1   /  1 + (Ω/Ω_c)^2N

Where;

H(Ω) = Magnitude of analog low pass filter

N = Order of the filter

Ω_c = Cut of frequency 

Silent features of low pass butter-worth filter :

• Since the magnitude response is nearly constant at lower frequencies. That means passband and are maximally flat.
• There are no ripples in the passband and also for stopband.
• The maximum gain occurs at the value at Ω = 0 and it is │H(0)│= 1
• The magnitude response is monotonically decreasing.

Application of Butter-worth filter :

• Butter-worth filter can be used as radar such as in designing the display of radar target track.
• In high quality, an audio application,s these are used.
• These are also used in the digital filter for motion analysis.
• This type of filter most commonly used in anti-aliasing filter in data converter applications

FIR filter block diagram

In the digital signal processing system, the use of  FIR short form is one type of filter whose impulse response is of finite duration, the reason of it settles zero in finite time. This is a contrast to IIR filter design, which has internal feedback and may continue to respond indefinitely. 

A discrete time FIR filter of N number of order and the top part is an N stage delay line with total N+1 taps to shown in the figure. Each of unit delay is a Z^-1 type of operator in the Z transform notation. 

The output y of a linear time-invariant system is determined by conveying its input signal x with its impulse response b. 

For a discrete-time FIR filter, the output is depend on a weighted sum of the current and finite number of previous values of the input signal.

The operation is described by the following equation, which defines the output sequence of y[n] in terms of its input sequence of x [n] given below.

Y[n] = b₀ x[n] + b₁ x[n-1]  + b₂ x[n-2] …………b_n x[n-N]

           N

Y[n] =  ∑ b_i x[n-i]

          I=0

Where, 

x[n] = input signal

y[n] = output signal

Advantages and disadvantages of FIR and IIR filters

FIR stands for finite impulse response. Most of the FIR full form filter is that linear phase filter When the linear phase filter is desired an FIR filter is usually used. Here this article gives information about the advantages and disadvantages of FIR as well as IIR filters to know more details about it. 

Advantages of  FIR filters :

• FIR filter is always stable
• It is simple
• Design complexity generally linear
• FIR filter are having linear phase response
• It is easy to optimize
• Non causal
• Transient have a finite duration
• Quantization noise is not much as a problem
• Round of noise error is minimum
• Both filtering recursive, as well as nonrecursive filter, can be also designed using FIR designing techniques
• For designing of a filter having an arbitrary magnitude response; FIR filter designing techniques can be easily applied
• Good performance
• Robust
• The necessity of computational techniques for filter implementation
• A requirement of large storage
• 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 to implement
• It is hard to implementation than IIR
• Expensive due to a large order
• Require more memory
• Time-consuming process

IIR stands for infinite impulse response. IIR Filter has a lower filter order then a corresponding FIR filter. Here this article also gives information about the advantages and disadvantages of  IIR filters to know more details about it.

Advantages of IIR filters :

• It has a stable design
• An IIR filter has a lesser number of side lobes in the stop band then FIR filter with the same number of parameters
• Implementation of an IIR filter involves fewer parameters 
• Less memory requirement
• Lower computational complexity
• Analog frequency and digital frequency are linearly related

Disadvantages of IIR filters :

• This filter is useful only when the analog filter is bandlimited
• They are harder to implement using fixed-point arithmetic, such as noise generated by calculations and also a limit cycles
• They are more susceptible to the problem of line finite length arithmetic
• They don't offer the computational advantages of FIR filter for the multi-rate application system

Explore more information:

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

IIR filter example

IIR filter short form is for infinite impulse response. In IIR filter design that are following criteria should be satisfied by designing the digital filter in the analog domain and transforming into a digital domain.

The transfer function of analog filter is :

H(s) = 3 / (s+2) (s+3) with Ts = 0.1 sec

Design the digital IIR filter using BLT.

Answer :

H(s) = 3 / (s+2) (s+3)........................(1)

In BLT H(Z) is obtained by putting :

s = 2/Ts [ Z-1/Z+1] here Ts = 0.1 sec

s = 2/0.1 [ Z-1/Z+1]  

s = 20 [ Z-1/Z+1]  

Putting this value in equation (1) we get,

H(Z) = 3 / [ 20*(Z-1/Z+1)+2] [  20*(Z-1/Z+1)+3]

H(Z) = 3 / [(20Z-20/Z+1) + 2] [ (20Z-20/Z+1 ) + 3 ]

H(Z) = 3 (Z+1) (Z+1) / (20Z-20+2Z+2) (20Z-20+3Z+3) 

H(Z) = 3 (Z+1)² /  506Z²-374Z-414Z+306

-

H(Z) = 3  (Z²+ 2Z +1 )/  506Z²-788Z+306

H(Z) =  (Z²+ 2Z +1 )/  168.67Z²-262.67Z+102

This function required a function for digital IIR filter 

IIR filter properties

IIR filter stands for infinite impulse response. In IIR filter design that are following criteria should be satisfied by designing a digital filter in the analog domain and transforming into the digital domain.

• IIR filter sometimes unstable
• IIR filter has difficult to design
• Lower filter order then a corresponding FIR filter
• Non zero pole exists in its transfer function
• Usually, have nonlinear property
• When phase distortion tolerable IIR usually favored
• IIR filter has lower computational
• IIR filter has less memory
• Less parameter to achieve a sharp cut off filter

FIR filter basics

Digital filter is very important in digital signal processing. As mentioned in the introduction, the filter has basically two uses: separation and signal restoration. One of the most popular filters knows as FIR filter, whose impulse response is of the finite period, as a result of it settles to zero in finite time. FIR filter is the most popular kind of filters can execute in software and these filter can be continuous time, analog or digital and discrete time.

What does FIR mean?

FIR filter stands for finite impulse response filter, It is a digital filter used in digital signal processing application. FIR filter can be easily designed to be in the linear phase.

In this filtering, if you put in an impulse, that is, a single "1" sample followed by many "0" samples, zeroes will come out after the "1" sample so has made its way through the delay line of the filter.

FIR filter is to simple to implement it is one type of advantages of FIR filters. One of the most digital signal processor the FIR calculation can be implemented, learn or done by looping a single instruction.

FIR filter sometimes has the disadvantages that they require more memory or calculation to achieve a given filter response characteristics.

Compared to IIR filter, FIR filter offers the following advantages :

• They are simple to implement, the FIR calculation can be done by looping a single instruction.
• They have desireable numeric properties. IIR filter can cause significant problems due to the use of feedback, but FIR filters without feedback can usually be implemented using fewer bits, and the designer has a fewer practical problem to solve related to non-ideal arithmetic.
• They can be implemented using fractional arithmetic.
• They can be easily be designed to be linear phase. 
• They are suited to multi-rate applications. 

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.

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 

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.