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  1. 1. Presented by Mr.Nilesh Rambahadur Jaiswar Mr.Kunal Dilip Vartak
  2. 2. INTRODUCTION In signal processing, the function of a filter is to remove unwanted parts of the signal, such as random noise, or to extract useful parts of the signal, such as the components lying within a certain frequency range.
  3. 3. Basic Setup Of Digital Filter
  4. 4. Operation Of Digital Filters : • Raw Signal : V = x(t) Where t is time. • This signal is sampled at time intervals h (the sampling interval). The sampled value at time t = ih is xi=x(ih) • Transferred From The ADC To The Processor x0 , x1 , x2 , x3 , ... • Corresponding to the values of the signal waveform at t = 0, h, 2h, 3h, ... • And t = 0 is the instant at which sampling begins. • At time t = nh the values available to the processor x0 , x1 , x2 , x3 , ... Xn • The digital output from the processor to the DAC y0 , y1 , y2 , y3 , ... yn
  5. 5. Examples Of Simple Digital Filters Unity Gain Filter:  yn = xn  Each output value yn is exactly the same as the corresponding input value xn: y0 = x0 , y1 = x1 , ................. Simple Gain Filter:  yn = Kxn  where K = constant.  This simply applies a gain factor K to each input value. Pure Delay Filter:  yn = xn-1  The output value at time t = nh is simply the input at time t = (n-1)h , i.e. the signal is delayed by time h : y0 = x-1 ,y1 = x0,.....
  6. 6. Two-term Difference Filter:  yn = xn– xn-1  The output value at t = nh is equal to the difference between the current input xn and the previous input xn-1 :  y0 = x0– x-1 ,  y1 = x1– x0 ,  y2 = x2– x1 , ... etc Two-term Average Filter:  yn = ( xn + xn-1 )/2  The output is the average (arithmetic mean) of the current and previous input:  y0 =(x0 + x-1)/2 ,  y1 =(x1 + x0)/2 ,  y2 =(x2 + x1)/2 , ... etc
  7. 7. Three-term Average Filter:  yn =(xn + xn-1 + xn-2)/ 3  This is similar to the previous example, with the average being taken of the current and two previous inputs:  y0 =(x0 + x-1 + x-2)/ 3  y1 =(x1 + x0 + x-1)/ 3  y2 =(x2 + x1 + x0)/ 3  As before, x-1 and x-2 are taken to be zero. Central Difference Filter:  yn =(xn– xn-2)/ 2  The output is equal to half the change in the input signal over the previous two sampling intervals:  y0 =(x0 – x-2)/ 2  y1 =(x1 – x-1)/ 2 , etc….
  8. 8. Types of Digital Filters  Finite Impulse Response , Or FIR Filters :  This filters express each output sample as a weighted sum of the last N input samples, where N is the order of the filter. FIR filters are normally non-recursive, meaning they do not use feedback and as such are inherently stable.  Infinite Impulse Response , Or IIR Filters:  This filters are the digital counter part to analog filters. Such a filter contains internal state, and the output and the next internal state are determined by a linear combination of the previous inputs and outputs
  9. 9. Finite impulse response  In signal processing , a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.  Definition
  10. 10. Properties  Are Inherently Stable  This is due to the fact that, because there is no required feedback, all the poles are located at the origin and thus are located within the unit circle .  Require No Feedback:  This means that any rounding errors are not compounded by summed iterations.  The same relative error occurs in each calculation.  This also makes implementation simpler .  They Can Easily Be Designed to be linear phase by making the coefficient sequence symmetric; linear phase, or phase change proportional to frequency, corresponds to equal delay at all frequencies.
  11. 11. Impulse response  The impulse response h[n]can be calculated if we set x[n]= δ[n]in the above relation, where δ[n] is the Kronecker delta impulse. The impulse response for an FIR filter then becomes the set of coefficients , as follows For n=0 to N  The Z-transform of the impulse response FIR filters are clearly bounded - input bounded-output (BIBO) stable, since the output is a sum of a finite number of finite multiples of the input values, so can be no greater than
  12. 12. Filter design To design a filter means to select the coefficients such that the system has specific characteristics. Window design method In the Window Design Method, one designs an ideal IIR filter, then applies a window function to it –Ve in the time domain, multiplying the infinite impulse by the window function. This results in the frequency response of the IIR being convolved with the frequency response of the window function. If the ideal response is sufficiently simple, such as rectangular, the result of the convolution can be relatively easy to determine. Frequency Sampling method Weighted least squares design
  13. 13. Moving Average Example Block diagram of a simple FIR filter (2nd-order/3-tap filter in this case, implementing a moving average)
  14. 14. Amplitude And Phase Responses
  15. 15.  A moving average filter is a very simple FIR filter. It is sometimes called a boxcar filter.  The filter coefficients, b0,......,bN are found via the following equation:  To provide a more specific example, we select the filter order:  The impulse response of the resulting filter is:  z-transform of the impulse response:
  16. 16. Infinite Impulse Response Infinite Impulse Response (IIR) is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters.  Implementation And Design IIR filters may be implemented as either analog or digital filters. In digital IIR filters, the output feedback is immediately apparent in the equations defining the output. Note that unlike FIR filters, in designing IIR filters it is necessary to carefully consider the "time zero" case in which the outputs of the filter have not yet been clearly defined.
  17. 17. Transfer Function Derivation  P is the feedforward filter order  bi are the feedforward filter coefficients  Q is the feedback filter order  ai are the feedback filter coefficients  x[n] is the input signal  y[n] is the output signal. When Rearranged
  18. 18.  IIR Filter Z Transfer Function  Description Of Simple IIR Filter Block Diagram
  19. 19. Stability The transfer function allows us to judge whether or not a system is bounded-input, bounded-output (BIBO) stable. To be specific, the BIBO stability criteria requires that the ROC(Radius Of Convergence) of the system includes the unit circle. For example, for a causal system, all poles of the transfer function have to have an absolute value smaller than one. In other words, all poles must be located within a unit circle in the Z-plane.  The poles are defined as the values Z of which make the denominator of H(z) equal to 0 :  If aj =0 then the poles are not located at the origin of the z-plane.
  20. 20. Applications of Digital filter  Communications Systems  Audio Systems Such As CD / Dvdplayers  Instrumentation  Image Processing And Enhancement  Processing Of Seismic And Other Geophysical Signals  Processing Of Biological Signals  Speech Synthesis
  21. 21. Advantages  It has linear phase response.  Thermal and environmental variation cannot change the performance.  It is possible to filter several input sequences without any hardware replication. Disadvantages  Actually the speed operation totally depends on the number of the arithmetic operation in the processor.  Finite word-length effect, which results quantizing noise and round-off noise.  It needs much longer time to design and develop the digital sequences though it can be used on other tasks or applications once developed.
  22. 22. Conclusion The main utility of the analysis methods presented is in ascertaining how a given filter will affect the spectrum of a signal passing through it. Some of the concepts introduced were linearity, time-invariance, filter impulse response, difference equations, transient response, steady-state response, transfer functions, amplitude response, phase response, phase delay, group delay, linear phase, minimum phase, maximum phase, poles and zeros, filter stability, and the general use of complex numbers to represent signals, spectra, and filters.
  23. 23. Under Guidance of Ms. Pradnya Vartak Presented by Mr. Nilesh Rambahadur Jaiswar Mr.Kunal Dilip Vartak