Chapter 6 e Digital Filter Structures
Chapter 6 Digital Filter Structures
86.1 Introduction The convolution sum description of an LTI discrete-time system can, in principle, be used to implement the system For an IIR finite-dimensional system this approach is not practical as here the impulse response is of infinite length However, a direct implementation of the IIR finite-dimensional system is practica
• The convolution sum description of an LTI discrete-time system can, in principle, be used to implement the system • For an IIR finite-dimensional system this approach is not practical as here the impulse response is of infinite length • However, a direct implementation of the IIR finite-dimensional system is practical §6.1 Introduction
86.1 Introduction Here the input-output relation involves a finite sum of products: yn]=-∑k=14kyn-+∑k20Pkxn- On the other hand, an FiR system can be implemented using the convolution sum e which is a finite sum of products y小]=∑0b1m-k1
• Here the input-output relation involves a finite sum of products: = = = − − + − M k k N k k y n d y n k p x n k 1 0 [ ] [ ] [ ] = = − N k y n h k x n k 0 [ ] [ ] [ ] §6.1 Introduction • On the other hand, an FIR system can be implemented using the convolution sum which is a finite sum of products:
86.1 Introduction The actual implementation of an LtI digital filter can be either in software or hardware form, depending on applications In either case, the signal varia bles and the filter coefficients cannot be represented with finite precision
• The actual implementation of an LTI digital filter can be either in software or hardware form, depending on applications • In either case, the signal variables and the filter coefficients cannot be represented with finite precision §6.1 Introduction
86.1 Introduction A structural representation using interconnected basic building blocks is the first step in the hardware or software implementation of an LTI digital filter The structural representation provides the key relations between some pertinent internal variables with the input and output that in turn provides the key to the implementation
• A structural representation using interconnected basic building blocks is the first step in the hardware or software implementation of an LTI digital filter • The structural representation provides the key relations between some pertinent internal variables with the input and output that in turn provides the key to the implementation §6.1 Introduction
§6,11 Block Diagram Representation In the time domain, the input-output relations of an lti digital filter is given by the convolution sum yn]=∑k=ohk]xn- or, by the linear constant coefficient difference equation y[n]=-ckerdkyvIn-k]+ 2ko pk[n-ki
§6.1.1 Block Diagram Representation • In the time domain, the input-output relations of an LTI digital filter is given by the convolution sum =− = − k y[n] h[k]x[n k] = = = − − + − M k k N k k y n d y n k p x n k 1 0 [ ] [ ] [ ] or, by the linear constant coefficient difference equation
§6,11 Block Diagram Representation For the implementation of an lti digital filter, the input-output relationship must be described by a valid computational algorithm To illustrate what we mean by a computational algorithm, consider the causal first-order lti digital filter shown below
§6.1.1 Block Diagram Representation • For the implementation of an LTI digital filter, the input-output relationship must be described by a valid computational algorithm • To illustrate what we mean by a computational algorithm, consider the causal first-order LTI digital filter shown below
§6,11 Block Diagram Representation The filter is described by the difference equation yIn]=-diyIn-1+poxIn+pixn-11 Using the above equation we can compute yIn for n20 knowing the initial condition yln-l and the input x[n for n≥-1
§6.1.1 Block Diagram Representation • The filter is described by the difference equation y[n]=-d1y[n-1]+p0x[n]+p1x[n-1] • Using the above equation we can compute y[n] for n0 knowing the initial condition y[n-1] and the input x[n] for n -1
§6,11 Block Diagram Representation y|0=-d1y{-1]+ poX+px -1 y[1=d1y|0|+p0X1+p1x[0 y[2]=-d1y[1]+poX[2]+p1x[1 We can continue this calculation for any value of the time index n we desire
§6.1.1 Block Diagram Representation y[0]=-d1y[-1]+p0x[0]+p1x[-1] y[1]=-d1y[0]+p0x[1]+p1x[0] y[2]=-d1y[1]+p0x[2]+p1x[1] .… • We can continue this calculation for any value of the time index n we desire
§6,11 Block Diagram Representation Each step of the calculation requires a knowledge of the previously calculated value of the output sample(delayed value of the output), the present value of the input sample, and the previous value of the input sample (delayed value of the input) As a result, the first-order difference equation can be interpreted as a valid computational algorithm
§6.1.1 Block Diagram Representation • Each step of the calculation requires a knowledge of the previously calculated value of the output sample (delayed value of the output), the present value of the input sample, and the previous value of the input sample (delayed value of the input) • As a result, the first-order difference equation can be interpreted as a valid computational algorithm