Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Filtering Periodic Signals

Navigation

Lenses

What is a lens?

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? tag icon

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

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Fundamentals of Electrical Engineering I"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice DSS - Braille display tagshide tags

    This module is included inLens: Rice University Disability Support Services's Lens
    By: Rice University Disability Support ServicesAs a part of collection: "Fundamentals of Electrical Engineering I"

    Comments:

    "Electrical Engineering Digital Processing Systems in Braille."

    Click the "Rice DSS - Braille" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Fundamentals of Electrical Engineering I"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "Fundamentals of Electrical Engineering I"

    Comments:

    "Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."

    Click the "Bookshare" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Fundamentals of Electrical Engineering I"

    Comments:

    "The course focuses on the creation, manipulation, transmission, and reception of information by electronic means. It covers elementary signal theory, time- and frequency-domain analysis, the […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

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.
 

Filtering Periodic Signals

Module by: Don Johnson. E-mail the author

Summary: This module demonstrates the effect that a time-invariant filter can have on periodic waveforms.

The Fourier series representation of a periodic signal makes it easy to determine how a linear, time-invariant filter reshapes such signals in general. The fundamental property of a linear system is that its input-output relation obeys superposition: L a 1 s 1 t+ a 2 s 2 t= a 1 L s 1 t+ a 2 L s 2 t L a 1 s 1 t a 2 s 2 t a 1 L s 1 t a 2 L s 2 t . Because the Fourier series represents a periodic signal as a linear combination of complex exponentials, we can exploit the superposition property. Furthermore, we found for linear circuits that their output to a complex exponential input is just the frequency response evaluated at the signal's frequency times the complex exponential. Said mathematically, if xt=ei2πktT x t 2 k t T , then the output yt=HkTei2πktT y t H k T 2 k t T because f=kT f k T . Thus, if xt x t is periodic thereby having a Fourier series, a linear circuit's output to this signal will be the superposition of the output to each component.

yt= k = c k HkTei2πktT y t k c k H k T 2 k t T
(1)
Thus, the output has a Fourier series, which means that it too is periodic. Its Fourier coefficients equal c k HkT c k H k T . To obtain the spectrum of the output, we simply multiply the input spectrum by the frequency response. The circuit modifies the magnitude and phase of each Fourier coefficient. Note especially that while the Fourier coefficients do not depend on the signal's period, the circuit's transfer function does depend on frequency, which means that the circuit's output will differ as the period varies.

Figure 1: A periodic pulse signal, such as shown on the left part ( ΔT=0.2 Δ T 0.2 ), serves as the input to an RC R C lowpass filter. The input's period was 1 ms (millisecond). The filter's cutoff frequency was set to the various values indicated in the top row, which display the output signal's spectrum and the filter's transfer function. The bottom row shows the output signal derived from the Fourier series coefficients shown in the top row.
Filtering a periodic signal
(a) Periodic pulse signal
Figure 1(a) (sig14.png)
(b) Top plots show the pulse signal's spectrum for various cutoff frequencies. Bottom plots show the filter's output signals.
Figure 1(b) (sig16.png)

Example 1

The periodic pulse signal shown on the left above serves as the input to a RC R C -circuit that has the transfer function (calculated elsewhere)

Hf=11+i2πfRC H f 1 1 2 f R C
(2)
Figure 1 shows the output changes as we vary the filter's cutoff frequency. Note how the signal's spectrum extends well above its fundamental frequency. Having a cutoff frequency ten times higher than the fundamental does perceptibly change the output waveform, rounding the leading and trailing edges. As the cutoff frequency decreases (center, then left), the rounding becomes more prominent, with the leftmost waveform showing a small ripple.

Exercise 1

What is the average value of each output waveform? The correct answer may surprise you.

Solution

Because the filter's gain at zero frequency equals one, the average output values equal the respective average input values.

This example also illustrates the impact a lowpass filter can have on a waveform. The simple RC R C filter used here has a rather gradual frequency response, which means that higher harmonics are smoothly suppressed. Later, we will describe filters that have much more rapidly varying frequency responses, allowing a much more dramatic selection of the input's Fourier coefficients.

More importantly, we have calculated the output of a circuit to a periodic input without writing, much less solving, the differential equation governing the circuit's behavior. Furthermore, we made these calculations entirely in the frequency domain. Using Fourier series, we can calculate how any linear circuit will respond to a periodic input.

Content actions

Download module as:

Add 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? tag icon

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