86. 1.1 Block Diagram Representation In the time domain, the input-output relations of an lti digital filter is given by the convolution sum y]=2∑k=小kn- or, by the linear constant coefficient difference equation y小=-∑:4yn-k]+2k20pm-k
§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
86. 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 yn
§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
86. 1.1 Block Diagram Representation The filter is described by the difference equation yIn=-diyln-1+poX/n+p,xn-1 Using the above equation we can compute yn for n20 knowing the initial condition yIn-1 and the input xn] for n>-1
§6.1.1 Block Diagram Representation • The filter is described by the difference equation y[n]=-d1 y[n-1]+p0 x[n]+p1 x[n-1] • Using the above equation we can compute y[n] for n≥0 knowing the initial condition y[n-1] and the input x[n] for n ≥-1
86. 1.1 Block Diagram Representation y|0=dy-]+px|0+p1x- yIl=-dy 0+poxl+p1X10 y2=-dyll+px2+px1 ●●● We can continue this calculation for any value of the time index n we d esire
§6.1.1 Block Diagram Representation y[0]=-d1y[-1]+p0x[0]+p1x[-1] y[1]=-d1 y[0]+p0 x[1]+p1 x[0] y[2]=-d1 y[1]+p0 x[2]+p1 x[1] .… • We can continue this calculation for any value of the time index n we desire
86. 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
§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