Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27
1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters 2 / 27
Definition The z-transform of a discrete-time signal x(n) is defined as the power series X(z) x(n)z n (1) n= where z is a complex variable, and hence, can be represented as a magnitude and phase. z = re jθ (2) It exists only for those values of z for which this series converges. 3 / 27
Summation Formula N a n = n=m { a M a N+1 1 a, if a 1 N M + 1, if a = 1 (3) 4 / 27
x(n) = a n u(n) Right-Sided Sequence Z X(z) = This definition is not complete without stating the region-of-convergence (ROC). 1 1 az 1 (4) Im(z) 1 0.8 a x(n) 0.6 0.4 0 Re(z) 0.2... 0-1 0 1 2 3 4 5 n... ROC 5 / 27
x(n) = a n u( n 1) left-sided Sequence Z X(z) = This definition is not complete without stating the region-of-convergence (ROC). 1 1 az 1 (5) Im(z) n -5-4 -3-2 -1 0 1 0... -0.1-0.2-0.3 x(n) 0 a Re(z) -0.4-0.5 ROC 6 / 27
Poles and Zeros If X(z) is rational function, then X(z) = B(z) A(z) = b 0 + b 1 z 1 + + b M z M a 0 + a 1 z 1 + + a N z N = b 0 a 0 z M+N (z z 1)(z z 2 ) (z z M ) (z p 1 )(z p 2 ) (z p N ) (6) z = z 1, z 2,...,z M and p = p 1, p 2,...,p N are the zeros and poles of X(z), respectively. 7 / 27
A relaxed linear time-invariant system with input x(n), an impulse response h(n), and an output y(n) can be expressed in the z-domain as Y(z) = H(z)X(z) (7) where H(z) is the transfer function of the system. 8 / 27
Difference Equation and hence, y(n) = N a k y(n k) + k=1 Y(z) X(z) = H(z) = M b k x(n k) (8) k=0 M k=0 b kz k 1 + N k=1 a kz k (9) 9 / 27
Causality and Stability Causality A system is causal if the ROC is the exterior of a circle. Stability A system is stable if the ROC includes the unit circle. Causal and Stable A system is both causal and stable if the poles are inside the unit circle. 10 / 27
Response to Complex Exponential Using the convolution equation y(n) = h(k)x(n k) (10) k= If this system is excited by a complex exponential x(n) = Ae jωn, < n < (11) where ω is an arbitrary frequency, then y(n) = AH(ω)e jωn (12) 11 / 27
Relationship to Fourier Transform If the ROC includes the unit circle, then H(ω) = H(z) z=e jω = h(n)e jωn (13) k= 12 / 27
Effect of Pole-Zero Location L p k ω z k Figure: Frequency response using the location of poles and zeros mag = product of distances from all zeros to L product of distances from all poles to L (14) 13 / 27
Filter Specs Figure: Filter Specifications 14 / 27
By Pole-Zero Placement Place zeros at locations you want the magnitude to decrease. Place Poles at locations you want the magnitude to increase. 15 / 27
Transformation from Analog Filters Use a transformation to convert analog filters to digital filters. The bilinear transformation is frequently used. s = 2 T ( ) 1 z 1 1 + z 1 (15) 16 / 27
Transformation from Analog Filters The bilinear transformation causes a frequency warping described by ω = 2 tan 1 ΩT 2 (16) 17 / 27
Transformation from Analog Filters Some common analog filter includes Butterworth: No ripple in either passband or stopband. Type I Chebyshev: ripple in passband only. Type II Chebyshev: ripple in stopband only. Elliptic: ripple in both passband and stopband. 18 / 27
Quantization Effects Due to quantization, the location of poles and zeros will be different than designed. This will cause the frequency response to be modified. To minimize the effects of quantization, build your filter as a cascade of second order systems. 19 / 27
A system using FIR filters has the output characterized by y(n) = M 1 k=0 h(k)x(n k) (17) FIR filters with symmetric and antisymmetric impulse responses have linear phase. 20 / 27
FIR Filter Design Using Windows 1 Compute the infinite impulse response from the an ideal impulse response. h d (ω) = 1 π H d (ω)e jωn dω (18) 2π π 2 Compute a finite impulse response using a finite length window. h(n) = h d (n)w(n) (19) 3 Windows include rectangular, Hamming, Hanning, Bartlet, Blackman, etc. 4 The wider the window, the sharper the cut-off. 21 / 27
. Figure: c Proakis and Manolakis, Digital Signal Processing, 4th Edition, Prentice Hall, 2007 22 / 27
. Figure: c Proakis and Manolakis, Digital Signal Processing, 4th Edition, Prentice Hall, 2007 23 / 27
FIR Filter Design Using Frequency-Sampling Method 1 Design the required filter in the Frequency domain. 2 Sample the designed magnitude response. 3 Add the linear phase. 4 Compute the IDFT to obtain h(n). 24 / 27
. 1.2 1.2 1 1 0.8 0.8 Magnitude 0.6 Magnitude 0.6 0.4 0.4 0.2 0.2 0 0 10 20 30 40 50 60 70 0 0 10 20 30 40 50 60 70 k k Figure: Frequency sampling approach with not transition points (left) and with transition points (right). 25 / 27
. 20 no transition points transition points 0-20 Magnitude -40-60 -80-100 -120 0 1 2 3 4 5 6 7 frequency Figure: Magnitude response using frequency sampling approach with not transition points (left) and with transition points (right). 26 / 27
FIR Filter Design Using Equiripple Method Uses the Alternation Theorem and solves an optimization problem resulting in an equal ripple. Design is accomplished by specifying bandedges and a weighting function. 27 / 27