Question

I. QUESTION A mapping that can be utilized to design a digital high-pass filter via an analog low-pass filter prototype is 1+z-1 1-2-1° S=- 1) Show that the imaginary axis in the s-plane maps to the unit circle in the 2-plane via this mapping. Hint. Use z = rejw and s = 0 + j12. 2) Show that the left-hand side of the imaginary axis in the s-plane maps to the interior of the unit circle in the z-plane via this mapping. 3) What is the relationship between the anaog frequency variable 12 and the digital frequency variable w implied by this mapping? Sketch the plot of 12 versus w. 4) Suppose you are interested in designing a high-pass digital filter with the following specifications: \H(w)| < 0.01, for Oslwlsh 0.95 s |H(w)| 5 1.05, for SWS A. Find the specifications of an analog low-pass filter that would generate this high-pass digital filter if the above mapping were to be used.

Answer 1

to find the Now, we have relation between Real part of 's' & 121 is not taken Now, (S=jus (= e(w/ Now It e j w is = for derivation It coswojsinw 1-1 cow-jsinw) js = 7 jis= (1 + coro) – j shrices (1- low) tjs inaw > jus = ((1+60,w) -jsimus) (01-60, w)-jsimw) Ci-Low)2 – (jeinwa ) juz = (1+60;w) (1–1040) – js inw (1 + lobo) + (1-465w)) - sinaw (It cosw-21030) - (- sinkw) - - ja = 1- cow aj sinw/2) - sinkw etsin²w t co²w - 2 los w. ja = 1 - 1 cow tsinw) - 2jsinw 141-2 low. ju= 1/- - Ojsinw - 2 sino 2 - 2 cow 2-2 cow r = - sinw - - & sin cos w/ T-cow 2 sim w/ . - » - costo - co+(00/2)

30, 2= - 10t/10/2). or w = cat "(-2) = n-Lot (2) w = 27- 2otle w Here, (wlen then only negative a) is allowed (Digital] frequently (NOT- Valid). (valid) A and on (Analog frequency) If, we try to plot lich then, lol = cot (w/). 10l<n (oina) - >w. 4) Given 14/W) Sool, oslw/ST/3. 0.95 < 1 HW)) <lios, 28610) LÃ Now, we know, 1) = cot (0/2) So w = 12) = cot (11/3/2) 11 ) = √3 = 1.732. & W=27/3 = cot (2/3/2) lal=0.5774

30/0.95 €/H (2) Cloos H() cool, oclole 0.5774. lal> 1.732