1 Conditions for Distortionless Transmission Transmission is said to be distortion less if the input and output have identical wave shapes within a multiplicative constant. Transmission is said to be distortion less if the input and output have identical wave shapes within a multiplicative constant. A delayed output that retains the input waveform is also considered distortion less. A delayed output that retains the input waveform is also considered distortion less. Thus in distortion-less transmission, the input x(t) and output y(t) satisfy the condition: y(t) = Kx(t - ) (1) where is the delay time and k is a constant. where is the delay time and k is a constant. Computing the Fourier Transform of (1) we obtain Computing the Fourier Transform of (1) we obtain Y(w) = KX(w)e -jw (2) Y(w) = KX(w)e -jw (2) The magnitude and phase response of (2) is given by The magnitude and phase response of (2) is given by
2 |H(w)| = K and (w) = -w = -2 f |H(w)| = K and (w) = -w = -2 f These are plotted in the following figure. K |H(w)| (w) ww -w Amplitude response Phase response A physical transmission system may have amplitude and phase responses such as those shown below: (w) |H(w)| w w
3 Ideal Filters Filter: A very general term denoting any system whose output is a specified function of its input. Frequency Selective Filters: Low-Pass, High-Pass, Band-Pass, Band-Stop. Ideal Low-Pass Filter: An ideal low-pass filter passes all Signal components having frequency less than w w radian per second with no distortion and completely attenuates signal components having frequencies greater than w c Hz. -w c w c w |H(w)| (w) w
4 Ideal High-Pass Filter: An ideal High-Pass filter passes all signal components greater than w w radian per second with no distortion and completely attenuates signal components having frequencies less than w w radian per second. |H(w)| -w c wcwc w w (w)
5 Ideal Band pass Filter: An ideal Band stop filter passes all signal components having frequencies in a band of B centered at the frequency w 0 with no distortion and completely attenuates signal components having frequencies outside this band. -w 0 w0w0 B
6 Ideal Band stop Filter: An ideal Band stop filter is defined in the following figure: |H(w)| (w) w w
7 Characteristics of Practical Frequency Selective Filters 1+ 1 1- 1 Passband ripple 1 = passband ripple 2 = Stopband ripple w c = w p = passband edge frequency. w s = stopband edge frequency. wpwp wsws 22
8 Analogue Filters: The Low-Pass Butterworth Approximation: The Low-Pass Butterworth Approximation: A Low-pass Butterworth filter has the amplitude response where n 1 is the filter order and the subscript b denotes the Butterworth filter. w c is the cutt-off frequency of the filter. (1) It is obvious from equation (1) that the Butterworth filter is an all Pole filter (i.e. N poles but no zeros).
w 0 1 N = 1 N=2 N=3 N=4 |H b (w)| The magnitude response of a Butterworth filter of order 1, 2, 3 and 4. Cutt-off Frequency is 1 radian per second.
10 or The poles of the filter are the roots of the denominator, i.e. or k = 0,1,2,…., N-1 The poles of a Butterworth filter can be computed as follows: From (1) (2)
11 Example1: Derive the transfer function of a first-order Butterworth filter. The cut-off frequency is 1 radian per second. Example1: Derive the transfer function of a first-order Butterworth filter. The cut-off frequency is 1 radian per second. Solution: The poles of a first-order Butterworth filter can be computed by putting k=0 and N = 1 in equation (2). i.e. s 0 = w c e j /2 e j /2 = e j (w c = 1) = cos + jsin = = -1 = cos + jsin = = -1 This means that the transfer function of the filter is
12 Example2: Repeat example 1 for a second order Butterworth filter. Example2: Repeat example 1 for a second order Butterworth filter. Solution: The poles of a second-order Butterworth filter can be computed by putting k=0, 1 and N = 2 in equation (2). i.e. s 0 = w c e j /2 e j /4 = e j 3 /4 (w c = 1) = cos(3 /4) + jsin(3 /4) = -1/ 2 + j1/ 2 = cos(3 /4) + jsin(3 /4) = -1/ 2 + j1/ 2 and s 1 = e j /2 e j3 /4 = e j5 /4 = -1/ 2 - j1/ 2 This means that the transfer function of the filter is Tutorial: Repeat example 2 for a 3 rd and 4 th order Butterworth filter.
13 Chebyshev Filter: There are two types of Chebyshev filters: Type1 Chebyshev Filters: These are all pole filters that Exhibit equi-ripple behaviour in the passband and a Monotonic characteristic in the stop band, as shown in the following figure. 0 1 wpwp w 1/(1+ 2 )
14 Type2 Chebyshev Filter: These filters contain both poles and zeros and exhibit a monotonic behaviour in the passband and an equiripple behaviour in the stopband. The magnitude response of a typical low-pass type 2 chebyshev filter is shown in the following figure
15 The magnitude of the frequency response characteristics of a type1 Chebyshev filter is given by where is a parameter of the filter that is related to the ripple in the pass-band and T N (x) is the Nth order Chebyshev polynomial defined as The Chebyshev polynomials can be generated by the recursive equation T N+1 (x) = 2xT N (x) – T N-1 (x), N = 1,2,… (3) where T 0 (x) = 1 and T 1 (x) = x. From (3) T 2 (x) = 2x 2 – 1, T 3 (x) = 4x 3 – 3x, and so on.
16 The filter parameter is related to the ripple in the passband, as shown in the figure of the previous slide. A relationship between passband ripple 1 and the parameter is given by A relationship between passband ripple 1 and the parameter is given by 1 = 10log(1 + 2 ) 1 = 10log(1 + 2 ) or = (10 1/10 – 1) or = (10 1/10 – 1) Example3: Derive transfer function of a first-order Cheby- Shev filter of type 1 with a unity gain and a passband ripple of 2dB. Solution: = (10 2/10 –1) = , T 1 2 = (w/w c ) 2
17 Therefore, Example 4: Find the transfer function for a second order normalized (w c = 1) Chebyshev low-pass filter with unity maximum gain and 1.5 dB of ripple in the passband. Solution: 1 = 1.5 dB, w c = 1, 2 = /10 – 1 =
18 Tutorial Q2: Derive the transfer function of a second order Low-pass chebyshev filter with unity dc gain and a passband Ripple of 2dB.