# Connexions

You are here: Home » Content » Noise bandwidth

### Recently Viewed

This feature requires Javascript to be enabled.

# Noise bandwidth

Module by: a.i. trivedi. E-mail the author

Summary: effects of mixing, integration and differentiation of noise are studied and noise bandwidth is defined

## Mixing of noise with a sinusoid

If kthkth size 12{k rSup { size 8{ ital "th"} } } {} component of noise is mixed with a sinusoid

n k t cos 2πf o t = a k 2 cos kΔf + f o t + b k 2 sin kΔf + f o t + a k 2 cos kΔf f o t + b k 2 sin kΔf + f o t n k t cos 2πf o t = a k 2 cos kΔf + f o t + b k 2 sin kΔf + f o t + a k 2 cos kΔf f o t + b k 2 sin kΔf + f o t alignl { stack { size 12{n rSub { size 8{k} } left (t right )"cos"2πf rSub { size 8{o} } t= { {a rSub { size 8{k} } } over {2} } "cos"2π left (kΔf+f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2π left (kΔf+f rSub { size 8{o} } right )t} {} # + { {a rSub { size 8{k} } } over {2} } "cos"2π left (kΔf - f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2π left (kΔf+f rSub { size 8{o} } right )t {} } } {}
(1)

Sum and difference frequency noise spectral components with 1/2 amplitude are generated and

G n f + f o = G n f f o = G n f 4 G n f + f o = G n f f o = G n f 4 size 12{G rSub { size 8{n} } left (f+f rSub { size 8{o} } right )=G rSub { size 8{n} } left (f - f rSub { size 8{o} } right )= { {G rSub { size 8{n} } left (f right )} over {4} } } {}
(2)

Considering power spectral components at kΔfkΔf size 12{kΔf} {} and lΔflΔf size 12{lΔf} {}, let the mixing frequency be f0=k+lΔff0=k+lΔf size 12{f rSub { size 8{0} } = { size 8{1} } wideslash { size 8{2} } left (k+l right )Δf} {}. This will generate 2 difference frequency components at the same frequency pΔf=f0kΔf=lΔff0pΔf=f0kΔf=lΔff0 size 12{pΔf=f rSub { size 8{0} } - kΔf=lΔf - f rSub { size 8{0} } } {}

Then difference frequency components are

n p1 t = a k 2 cos 2πpΔ ft b k 2 sin 2πpΔ ft n p1 t = a k 2 cos 2πpΔ ft b k 2 sin 2πpΔ ft size 12{n rSub { size 8{p1} } left (t right )= { {a rSub { size 8{k} } } over {2} } "cos"2πpΔ ital "ft" - { {b rSub { size 8{k} } } over {2} } "sin"2πpΔ ital "ft"} {}
(3)
n p2 t = a l 2 cos 2πpΔ ft + b l 2 sin 2πpΔ ft n p2 t = a l 2 cos 2πpΔ ft + b l 2 sin 2πpΔ ft size 12{n rSub { size 8{p2} } left (t right )= { {a rSub { size 8{l} } } over {2} } "cos"2πpΔ ital "ft"+ { {b rSub { size 8{l} } } over {2} } "sin"2πpΔ ital "ft"} {}
(4)

But as akal¯=akbl¯=bkal¯=bkbl¯=0akal¯=akbl¯=bkal¯=bkbl¯=0 size 12{ {overline {a rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {a rSub { size 8{k} } b rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } b rSub { size 8{l} } }} =0} {}

We find

E n p1 t n p2 t = 0 E n p1 t n p2 t = 0 size 12{E left [n rSub { size 8{p1} } left (t right )n rSub { size 8{p2} } left (t right ) right ]=0} {}
(5)

And

E n p1 t + n p2 t 2 = E n p1 t 2 + E n p2 t 2 E n p1 t + n p2 t 2 = E n p1 t 2 + E n p2 t 2 size 12{E left lbrace left [n rSub { size 8{p1} } left (t right )+n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace =E left lbrace left [n rSub { size 8{p1} } left (t right ) right ] rSup { size 8{2} } right rbrace +E left lbrace left [n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace } {}
(6)

Thus superposition of power applies even after shifting due to mixing.

## Minimizing Noise in Systems by filtering:

Assume white noise with Gnf=η2Gnf=η2 size 12{G rSub { size 8{n} } left (f right )= { {η} over {2} } } {}

To minimize noise entering the demodulator, a filter of bandwidth B can be placed, with B just wide enough to pass signal of interest. Output noise depends on the filter used.

Ideal LPF with white noise has No=ηBNo=ηB size 12{N rSub { size 8{o} } =ηB} {}

rectangular BPF with white noise has No=2η2f2f1=ηf2f1No=2η2f2f1=ηf2f1 size 12{N rSub { size 8{o} } =2 { {η} over {2} } left (f rSub { size 8{2} } - f rSub { size 8{1} } right )=η left (f rSub { size 8{2} } - f rSub { size 8{1} } right )} {}

### RC Low Pass Filter:

The filter transfer function is

H f = 1 1 + j f f c H f = 1 1 + j f f c size 12{H left (f right )= { {1} over {1+j { {f} over {f rSub { size 8{c} } } } } } } {}
(7)

Using Gnof=Hf2GnifGnof=Hf2Gnif size 12{G rSub { size 8{ ital "no"} } left (f right )= lline H left (f right ) rline rSup { size 8{2} } G rSub { size 8{ ital "ni"} } left (f right )} {}

we have

G no f = η 2 1 1 + f f c 2 G no f = η 2 1 1 + f f c 2 size 12{G rSub { size 8{ ital "no"} } left (f right )= { {η} over {2} } { {1} over {1+ left ( { {f} over {f rSub { size 8{c} } } } right ) rSup { size 8{2} } } } } {}
(8)

and noise power at filter o/p is

N o = G n f df = η 2 df 1 + f f c 2 N o = G n f df = η 2 df 1 + f f c 2 size 12{N rSub { size 8{o} } = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {G rSub { size 8{n} } left (f right ) ital "df"= { {η} over {2} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { { { ital "df"} over {1+ left ( { {f} over {f rSub { size 8{c} } } } right ) rSup { size 8{2} } } } } } } {}
(9)

using x=ffcx=ffc size 12{x= { {f} over {f rSub { size 8{c} } } } } {} and noting that dx1+x2=πdx1+x2=π size 12{ Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { { { ital "dx"} over {1+x rSup { size 8{2} } } } } =π} {}, we have

N o = π 2 ηf c N o = π 2 ηf c size 12{N rSub { size 8{o} } = { {π} over {2} } ηf rSub { size 8{c} } } {}
(10)

## Differentiating Filter:

The transfer function is

H f = j2 πτ f H f = j2 πτ f size 12{H left (f right )=j2 ital "πτ"f} {}
(11)

white noise creates output psd

G no f = 2 τ 2 f 2 η 2 G no f = 2 τ 2 f 2 η 2 size 12{G rSub { size 8{ ital "no"} } left (f right )=4π rSup { size 8{2} } τ rSup { size 8{2} } f rSup { size 8{2} } { {η} over {2} } } {}
(12)

and following this by a rectangular lpf of bandwidth B, noise at o/p is

No=BBGnofdf=23ητ2B3No=BBGnofdf=23ητ2B3 size 12{N rSub { size 8{o} } = Int cSub { size 8{ - B} } cSup { size 8{B} } {G rSub { size 8{ ital "no"} } left (f right ) ital "df"= { {4π rSup { size 8{2} } } over {3} } ital "ητ" rSup { size 8{2} } B rSup { size 8{3} } } } {}

## Integrating Filter:

An integrator integrating over an interval T has transfer function

H f = 1 j ωτ e jωT j ωτ H f = 1 j ωτ e jωT j ωτ size 12{H left (f right )= { {1} over {j ital "ωτ"} } - { {e rSup { size 8{ - jωT} } } over {j ital "ωτ"} } } {}
(13)

and thus

H f 2 = T τ 2 sin π Tf π Tf 2 H f 2 = T τ 2 sin π Tf π Tf 2 size 12{ lline H left (f right ) rline rSup { size 8{2} } = left ( { {T} over {τ} } right ) rSup { size 8{2} } left ( { {"sin"π ital "Tf"} over {π ital "Tf"} } right ) rSup { size 8{2} } } {}
(14)

The noise power o/p with white noise input

N o = η 2 T τ 2 sin π Tf π Tf 2 df = ηT 2 N o = η 2 T τ 2 sin π Tf π Tf 2 df = ηT 2 size 12{N rSub { size 8{o} } = { {η} over {2} } left ( { {T} over {τ} } right ) rSup { size 8{2} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { left ( { {"sin"π ital "Tf"} over {π ital "Tf"} } right ) rSup { size 8{2} } ital "df"= { {ηT} over {2τ rSup { size 8{2} } } } } } {}
(15)

(The integral has a value = π)

## Noise Bandwidth:

If a real filter with transfer fn HfHf size 12{H left (f right )} {} centered at fofo size 12{f rSub { size 8{o} } } {} is used, we can consider an equivalent rectangular filter centered at fofo size 12{f rSub { size 8{o} } } {} with a bandwidth BNBN size 12{B rSub { size 8{N} } } {} passing the same noise power.

• BNBN size 12{B rSub { size 8{N} } } {} is called the noise bandwidth of the real filter

• For the RC Filtered curves, the area can be shown to be
N o RC = π 2 ηf c N o RC = π 2 ηf c size 12{N rSub { size 8{o} } left ( ital "RC" right )= { {π} over {2} } ηf rSub { size 8{c} } } {}
(16)
• For a rectangular filter we have
N o rect = η 2 2B N = ηB N N o rect = η 2 2B N = ηB N size 12{N rSub { size 8{o} } left ( ital "rect" right )= { {η} over {2} } 2B rSub { size 8{N} } =ηB rSub { size 8{N} } } {}
(17)
• Setting N0(RC)=N0(rect)N0(RC)=N0(rect) size 12{N rSub { size 8{0} } $$ital "RC"$$ =N rSub { size 8{0} } $$ital "rect"$$ } {}, BN=π2fcBN=π2fc size 12{B rSub { size 8{N} } = { {π} over {2} } f rSub { size 8{c} } } {}

Hence noise BW of RC filter is 1.57 times its 3 dB BW.

## 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.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

### 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?

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