2 Digital design of Chebyshev filterBackground. The Type-1 Chebyshev filter is known to provide the...

Question

2  Digital design of Chebyshev filterBackground. The Type-1 Chebyshev filter is known to provide the lowest deviation in passband for a given filter order. The transfer function for the second order analogue low pass 3-dB passband ripple Chebyshev-1 filter is given by the following expressionwhere ?c = 2pfc and fc is a cut off frequency.A digital implementation of this analogue prototype is required for cut-off frequencyfc= 1 kHz. You are to perform the following tasks 2.A–2.D :2.A. Develop a digital implementation of the given analogue prototype by impulse invariant method.[20/100] marks2.B. Develop a digital implementation of the given analogue prototype by bi-linear transform method. Draw the schematic diagrams corresponding to both solutions
	
Document Preview: 2 	Digital design of  Chebyshev filterBackground.  The Type-1 Chebyshev filter  is known to provide the lowest deviation in passband for a given  filter  order. The  transfer function for the second order analogue low pass 3-dB passband ripple Chebyshev-1 filter  is given  by the following expressionwhere ?c  = 2pfc  and fc  is a cut off frequency.A digital implementation of this analogue prototype is  required for  cut-off frequencyfc  = 1 kHz.  You are to perform the following tasks 2.A–2.D :2.A. Develop a digital implementation of the given  analogue prototype by  impulse invariant method.[20/100] marks2.B. Develop a digital implementation of the given  analogue prototype by  bi-linear transform method.Created with an evaluation copy of Aspose.Words. To discover the full versions of our APIs please visit: https://products.aspose.com/words/Draw the schematic diagrams corresponding  to both solutions[20/100] marksCreated with an evaluation copy of Aspose.Words. To discover the full versions of our APIs please visit: https://products.aspose.com/words/2.C. Plot  the frequency transfer functions H (ej? ) for both solutions 2.A and 2.B  for a couple reasonable sample rates of your choice. Comment on  the differences.[20/100] marksCreated with an evaluation copy of Aspose.Words. To discover the full versions of our APIs please visit: https://products.aspose.com/words/2.D. Compare the results of the solutions 2.A and 2.B.[10/100] marksCreated with an evaluation copy of Aspose.Words. To discover the full versions of our APIs please visit: https://products.aspose.com/words/SubmissionThe following documents must be submitted electronically via Blackboard submission system:• Handwritten and scanned, or  electronic notes describing solutions to  the above problems.Please make sure that in case of multiple or repeated submission the last submission is complete as you  want it to be  submitted.AssignmentThis Assignment will...

Robert · Accepted Answer

2.A.  Given,
Assume the sampling frequency    as 10 kHz 
Therefore   
 
  
       
the equation can also be written as, 
 ( )  
  
 
                   
 
    ( )  
(    ) 
                   (    ) 
 
 ( )  
        
    (   
       
    
  
        
    
)
 
 ( )  
           
                       
 
    ( )  
           
(         ) 
 
From the properties of the impulse invariant transformations,
(    ) 
 
(  ) 
(   )
     
[
 
         
]       
Here      The above equation becomes,
(    ) 
 
(  ) 
( )
  
[
 
         
]            
   
        
(         ) 
           
The system function for digital filter thus obtained is  
 ( )  
                    
(         ) 
           
    ( )  
                          
(               ) 
 
Assuming T = .0001sec, the system function becomes 
 ( )  
              
                  

                   
 
Such a large gain is the characteristics of the IIR filter. To keep the gain down (to avoid the 
overflow when the filter is implemented) it is a common practice to divide the gain by   , the 
new transfer function becomes,
                   
 
1.637
-.6703
.2290
Z
-1
Z
-1
x[n]
y[n]

2.B. From the bilinear transformation, we have 
 ( )   ( ) 
  
 (   )
 (   )
 
Taking         ,

2 Digital design of Chebyshev filter Background. The Type-1 Chebyshev filter is known to provide the lowest deviation in passband for a given filter order. The transfer function for the second order...

Solution

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment