INC 212 Signals and systems Lecture#8: Analog filter design Assoc. - PowerPoint PPT Presentation

INC 212 Signals and systems Lecture#8: Analog filter design Assoc. Prof. Benjamas Panomruttanarug benjamas.pan@kmutt.ac.th

Filter x ( t ), X ( j  ) y ( t ) = x ( t )  h ( t ) LTI system, h ( t ), H ( j  ) Y ( j  ) = X ( j  ) H ( j  ) • Filter ≡ system • Can be either continuous or discrete time • By “distortionless transmission” we mean that the output signal of the system is an exact replica of the input signal, except, possibly, for two minor modifications: 1) A scaling of amplitude 2) A constant time delay 2 BP INC212

Time ‐ domain condition for distortionless transmission of a signal through an LTI system. A signal x ( t ) is transmitted through the system without distortion if the output signal y ( t ) is defined by    y ( t ) Cx ( t t )   j ω t Y(j ) CX(j ω e ) 0 0 where constant C accounts for a change in amplitude and constant t 0 accounts for a delay in transmission. Y(j ω )       j ω t h ( t ) C ( t t ) H(j ω ) Ce 0 0 X(j ω ) 3 BP INC212

8.3 Ideal Low ‐ Pass Filters  The frequency response of a filter is characterized by a passband and a stopband, which are separated by a transition band. p   ; v   4 BP INC212

8.3 Ideal Low ‐ Pass Filters (cont.)       j t e 0 ,     c H j ( ) ,    0,   c 5 BP INC212

Practical low ‐ pass filter: the passband, transition band, and stopband are shown for positive frequencies. Maximum passband attenuation or ripple in passband d   ; e   ; v   Maximum stopband magnitude or stopband attenuation Passband edge frequency Stopband edge frequency 6 BP INC212

8.4 Design of Filters Analog filter – Butterworth filter – Elliptic filter – Chebyshev filter (type I and II) 7 BP INC212

Analog filter • Butterworth filter(butter): – Maximally flat magnitude response – Number of poles = number of filter’s order – No zero • Elliptic filter (ellip): – Equiripple magnitude response on both bands • Chebyshev filter: – Type I (cheby1): ripples on passband – Type II (cheby2): ripples on stopband 8 BP INC212

Example Design analog Butterworth, Elliptic, and Chebyshev type I and II filters satisfying the following conditions: – Low ‐ pass filter with a cutoff at 1 rad/sec – Order of filter = 10 – Maximum ripple in the passband is 6 dB – Stopband attenuation is 50 dB (Hint: use Matlab command ‘bode’ to plot the frequency responses of the filters) 9 BP INC212

Practical use of an analog filter Low ‐ pass Butterworth filters driven from ideal current source: (a) order K = 1 and (b) order K = 3. Analog filter can be built from a passive filter!!! 10 BP INC212

Example Use the designed filters to filter the signal below:   s ( t ) 1 . 2 cos( 0 . 3 t ) 0 . 7 sin ( 5 t )  t  0 0 . 99 11 BP INC212