jwRC jwc R jwc Vi Vo wH + = + = = 1 1 /1 /1 )( 2 1 ) (1 1 |)(| = +

```Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Third Year
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
Analog Filters Design
Analog filter is a circuit which has been designed to pass signals with desired
frequencies and reject or attenuate others.
Passive Filter: Is a filter that consists of only passive elements such as (resistors,
capacitors, inductors).
Active Filter: Is a filter that consists of active elements such as (transistors, operation
amplifiers) as well as passive elements.
Generally, there are four types of Analog Filters whether passive or Active.
Passive Filters:
1. LOW-PASS Filters: Passes the low range of frequency and cut-off high
frequencies: the typical low-pass filter circuit is shown below:
C
H ( w) =
Vo
1 / jwc
Vo
1
=
=
Vi R + 1 / jwc 1 + jwRC Where s=jw
H(0)= 1, H(∞)=0
Note: We can calculate Wc by putting |H(w)|= 1 /
2 and then find the W where it
equal to Wc.
| H (Wc) |=
1
1 + (WcRC) 2
=
1
1
Wc
=
2 RC
(1)
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Third Year
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
|H(w)|
Actual
Ideal
0
W
Wc
Frequency response of Low-Pass filter
2. HIGH-PASS Filters: Passes the High range of frequency and cut-off low
frequencies as shown:
C
Vo
H ( w) =
Vo
R
jwRC
=
=
Vi R + 1 / jwc 1 + jwRC
H(0)= 0, H(∞)=1
|H(w)|
Ideal
1
Wc =
RC
0
Wc
Actual
W
Frequency response of High-Pass filter
(2)
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Third Year
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
3. BAND-PASS Filters: Passes range of frequency within band and block other
frequencies outside that band as shown:
L
C
Vo
H ( w) =
Vo
R
=
Vi R + j ( wL − 1 / wC )
H(0)= 0,
The center frequency which is given by:
Wo =
1
LC
When |H(Wo)|=1
and
|H(w)|
W1&lt; Wo &lt; W2
Ideal
1
0.707
0
Actual
Wc1
Wc2
W
Wo
(3)
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Third Year
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
4. BAND-STOP Filters: Passes range of frequency out side the frequency band and
block other frequencies within that band as shown:
R
L
Vo
DC
C
H ( w) =
Vo
j ( wL − 1 / wC )
=
Vi R + j ( wL − 1 / wC )
H(0)= 1,
H(∞)=1.
The center frequency which is given by:
Wo =
1
LC
and
W1&lt; Wo &lt; W2
|H(w)| Wo
1
0.707
0
Ideal
Actual
Wc1 Wc2
W
(4)
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Third Year
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
Table 1: Ideal Characteristics of the four type of filters (first and seconds orders).
Filter Type
H(0)
H(Wc) or H(Wo)
1
H(∞)
0
Low-pass
High-pass
0
1
1/ 2
Band-pass
0
0
1
Band-stop
1
1
0
1/ 2
Notes:
1. Wc: is the cut off frequency for low-pass and high-pass filters.
2. Wo: is the center frequency for band-lass and band-stop filters.
3. That mean, we can use this table to know the type of filter (or the type of the
transfer function whether first or second order).
Example: Determine what type of filter shown in the figure below, take R=2k,
L=2H and C=2&micro;F.
L
Vi
C
Vo
(5)
Third Year
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
Ans:
(6)
Third Year
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
H.W: what type of filter of the following circuit? Where R1=100 ohm, R2=100 ohm
and L=2 mH.
R1
Vi
L
R2
(7)
Third Year
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
Active Filters:
A- Low-Pass Active Filter:
b- High-Pass Active Filter:
(8)
Third Year
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
H.W: Obtain the transfer function of the following active filter, what type is it?
H.W: Find the transfers function of the following active filter and what kind is it?
Nyquist–Shannon sampling theorem:
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon,
is
a
fundamental
result
in
the
field
of
information
theory,
in
particular
telecommunications and signal processing. Sampling is the process of converting a
signal (for example, a function of continuous time or space) into a numeric sequence (a
function of discrete time or space). Shannon's version of the theorem states:[1]
If a function x(t) contains no frequencies higher than B hertz, it is completely
determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
(9)
Third Year
Diyala University - College of Engineering
Computer &amp; Software Engineering Department
Digital Signal Processing
‫بسم ﷲ الرحمن الرحيم‬
Lecture 15
---------------------------------------------------------------------------------------------------------------------------
Example: Consider the analog signal:
Xa(t)= 3 cos 100π t.
1. Determine the minimum sampling rate required to avoid the aliasing.
2. Suppose that the signal is sampled at the rate Fs=200 Hz. What is the discretetime signal obtained after sampling?