MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.1 0 W j W s x x x x x oo 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 61 Reading: • Class handout: Sinusoidal Frequency Response of Linear Systems (Sec. 7). • Class handout: Sinusoidal Frequency Response of Linear Systems (Sec. 6.1). • Class handout: Introduction to Continuous Time Filter Design. Poles and Zeros of Filter Classes We can use the s-plane relationship between a filter’s poles and zeros and the frequency response function to make some general comments about the desgn of various classes of filters: (a) Low-Pass Filters: • To ensure a high-frequency roll-off the number of poles must exceed the number of zeros, ie n>m. (In many low-pass filters m = 0.) • To ensure a finite low frequency gain there can be no poles or zeros at the origin. s - p l a n e| H ( j W ) | n > m n o p o l e s o r z e r o s a t t h e o r i g i n (b) High-Pass Filters • To ensure a constant high-frequency gain the number of poles must equal the number of zeros, ie n = m. • To ensure a low frequency gain that approaches zero, there must be one or more zeros at the origin. 1copyright cD.Rowell 2008 6–11 0 W j W s W c x o o o x x 1 0 W s o o o x x x x x x s - p l a n e| H ( j W ) | n = m n z e r o s a t t h e o r i g i n (c) Band-Pass Filters • To ensure a high-frequency roll-off the number of poles must exceed the number of zeros, ie n>m. • To ensure a low frequency gain that approaches zero, there must be one or more zeros at the origin. • The band-pass characteristic is shaped by a group of poles clustered near the imaginary axis in the region of the passband, j W s - p l a n e| H ( j W ) | n > m p o l e s c l u s t e r e d n e a r t h e i m a g i n a r y a x i s i n r e g i o n o f t h e p a s s - p a n d o n e o r m o r e z e r o s a t t h e o r i g i n W c1 W c2 (d) Band-Stop Filters • To ensure a constant high-frequency gain the number of poles must equal the number of zeros, ie n = m. • To ensure a finite low frequency gain there can be no poles or zeros at the origin. • The band-reject characteristic is shaped by a group of zeros clustered on or near the imaginary axis in the region of the stopband, 6–21 0 W j W s W r W r 1 2 o o o o o o x x x x x x � � � � 2 s - p l a n e n = m | H ( j W ) | z e r o s c l u s t e r e d o n o r n e a r t h e i m a g i n a r y a x i s i n r e g i o n o f t h e s t o p - p a n d n o p o l e s o r z e r o s a t t h e o r i g i n The Decibel Filter frequency response magnitudes |H(jΩ)| are frequently plotted using the decibel log-arithmic scale. The Bel, named after Alexander Graham Bell, is defined as the logarithm to base 10 of the ratio of two power levels. In practice the Bel is too large a unit, and the decibel (abbreviated dB), defined to be one tenth of a Bel, has become the standard unit of logarithmic power ratio. The power flow P into any element in a system, may be expressed in terms of a logarithmic ratio Q to a reference power level Pref : P P Q = log10 Bel or Q = 10 log10 dB. (1)Pref Pref Because the power dissipated in a D–type element is proportional to the square of the amplitude of a system variable applied to it, when the ratio of across or through variables is computed the definition becomes � �2 � � A A Q = 10 log10 = 20 log10 dB. (2)Aref Aref where A and Aref are amplitudes of variables. 2 Table ?? expresses some commonly used decibel values in terms of the power and amplitude ratios. The magnitude of the frequency response function |H (jΩ)| is defined as the ratio of the amplitude of a sinusoidal output variable to the amplitude of a sinusoidal input variable. This ratio is expressed in decibels, that is |Y (jΩ)|20 log10 |H(jΩ)| = 20 log10 |U(jΩ)| dB. As noted this usage is not strictly correct because the frequency response function does not define a power ratio, and the decibel is a dimensionless unit whereas |H (jΩ)| may have physical units. 2This definition is only strictly correct when the two amplitude quantities are measured across a common D–type (dissipative) element. Through common usage, however, the decibel has been effectively redefined to be simply a convenient logarithmic measure of amplitude ratio of any two variables. This practice is widespread in texts and references on system dynamics and control system theory. In this book we have also adopted this convention. 6–30 1 1 1 3 Decibels Power Ratio Amplitude Ratio -40 0.0001 0.01 -20 0.01 0.1 -10 0.1 0.3162 -6 0.25 0.5 -3 0.5 0.7071 0 1.0 1.0 3 2.0 1.414 6 4.0 2.0 10 10.0 3.162 20 100.0 10.0 40 10000.0 100.0 Table 1: Common Decibel quantities and their corresponding power and amplitude ratios. Low-Pass Filter Design The prototype low-pass filter is based upon the magnitude-squared of the frequency response function |H(jΩ)|2, or the frequency response power function. The phase response of the filter is not considered. We begin by defining tolerance regions on the power frequency response plot, as shown below:. | H ( j W ) |2 1 + e 2 1 + l 2 0 W c W r W ( r a d / s e c ) p a s s b a n d s t o p b a n dt r a n s i t i o n b a n d The filter specifications are that 1 ≥|H(jΩ)|2 > 1 for |Ω| < Ωc1+ 2 1 and |H(jΩ)|2 < for |Ω| > Ωr,1+ λ2 where …