# Connexions

You are here: Home » Content » Signals, Systems, and Society » First Order Filters

• #### 4. The Laplace Transform

• 5. References for Signals, Systems, and Society

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Textbook):

Textbook by: Carlos E. Davila. E-mail the author

# First Order Filters

Module by: Carlos E. Davila. E-mail the author

A first-order lowpass filter has the frequency response

H L P ( j Ω ) = 1 1 + j Ω Ω c H L P ( j Ω ) = 1 1 + j Ω Ω c
(1)

The frequency at which the frequency response magnitude has dropped to 1/21/2 is called the corner frequency1. The frequency response magnitude and phase are plotted in Figure 1. It is common to express the frequency response magnitude in units of decibels (dB) using the formula

20 l o g 10 H ( j Ω ) 20 l o g 10 H ( j Ω )
(2)

At the corner frequency for a first order lowpass filter, the frequency response magnitude is 1/21/2 or roughly -3 dB. From (Reference), it can easily be seen that the impulse response for the first-order lowpass filter is given by

h L P ( t ) = Ω c e - Ω c t u ( t ) h L P ( t ) = Ω c e - Ω c t u ( t )
(3)

A first-order highpass filter is given by

H H P ( j Ω ) = j Ω Ω c 1 + j Ω Ω c H H P ( j Ω ) = j Ω Ω c 1 + j Ω Ω c
(4)

Notice that

H H P ( j Ω ) = 1 - 1 1 + j Ω Ω c H H P ( j Ω ) = 1 - 1 1 + j Ω Ω c
(5)

This makes sense since a highpass filter can be constructed by taking the filter input x(t)x(t) and subtracting from it a lowpass filtered version of x(t)x(t). The impulse response of the first-order highpass filter therefore becomes:

h H P ( t ) = δ ( t ) - Ω c e - Ω c t u ( t ) h H P ( t ) = δ ( t ) - Ω c e - Ω c t u ( t )
(6)

First order filters can be easily implemented using linear circuit elements like resistors, capacitors, and inductors. Figure 3 shows a first order filter based on a resistor and a capacitor. Since the impedance for a resistor and capacitor are RR and 1/jΩC1/jΩC, respectively, voltage division leads to a frequency response of

H L P ( j Ω ) = 1 1 + j Ω R 1 C 1 H L P ( j Ω ) = 1 1 + j Ω R 1 C 1
(7)

Therefore the corner frequency for this filter is Ωc=1R1C1Ωc=1R1C1. Similarly, a first-order highpass filter can be implemented using a resistor and capacitor as shown in Figure 4. This filter has a frequency response of

H H P ( j Ω ) = j Ω R 2 C 2 1 + j Ω R 2 C 2 H H P ( j Ω ) = j Ω R 2 C 2 1 + j Ω R 2 C 2
(8)

The corner frequency for the highpass filter is seen to be Ωc=1R2C2Ωc=1R2C2.

Now one might be tempted to apply the results of (Reference) to build a bandpass filter by cascading the lowhpass and highpass circuits in Figures Figure 3 and Figure 4, respectively. Theory would predict that the equivalent frequency response of this circuit is given by

H e q ( j Ω ) = H L P ( j Ω ) H H P ( j Ω ) H e q ( j Ω ) = H L P ( j Ω ) H H P ( j Ω )
(9)

Unfortunately, this is not possible since the circuit elements in the lowpass and highpass filters interact with one another and therefore affect the overall behavior of the circuit. This interaction between the two circuits is called loading will be studied in greater detail in the exercises. To get theoretical behavior, it is necessary to use a voltage follower circuit, between the lowpass filter from the highpass circuits. The voltage follower circuit is usually an active circuit (requires external power supply) that has very high input impedance and very low output impedance. This eliminates any loading effects which would normally occur between the lowpass and highpass filter circuits.

## Footnotes

1. This term most likely originates from it's role in Bode plots, a shortcut method for sketching the graph of a frequency response.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks