Mixing of noise with a sinusoid If kth 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 2π kΔf + f o t + b k 2 sin 2π kΔf + f o t + a k 2 cos 2π kΔf − f o t + b k 2 sin 2π 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 {} } } {} 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 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} } } {} Considering power spectral components at kΔf size 12{kΔf} {} and lΔf size 12{lΔf} {}, let the mixing frequency be f0=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=f0−kΔf=lΔf−f0 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 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"} {} 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"} {} But as akal¯=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 size 12{E left [n rSub { size 8{p1} } left (t right )n rSub { size 8{p2} } left (t right ) right ]=0} {} And 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 } {} Thus superposition of power applies even after shifting due to mixing.

Minimizing Noise in Systems by filtering:

Assume white noise with Gnf=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=ηB size 12{N rSub { size 8{o} } =ηB} {} rectangular BPF with white noise has No=2η2f2−f1=ηf2−f1 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 size 12{H left (f right )= { {1} over {1+j { {f} over {f rSub { size 8{c} } } } } } } {} Using Gnof=∣Hf∣2Gnif 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 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} } } } } {} and noise power at filter o/p is 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} } } } } } } {} using x=ffc size 12{x= { {f} over {f rSub { size 8{c} } } } } {} and noting that ∫−∞∞dx1+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 size 12{N rSub { size 8{o} } = { {π} over {2} } ηf rSub { size 8{c} } } {}

Differentiating Filter: The transfer function is H f = j2 πτ f size 12{H left (f right )=j2 ital "πτ"f} {} white noise creates output psd G no f = 4π 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} } } {} and following this by a rectangular lpf of bandwidth B, noise at o/p is No=∫−BBGnofdf=4π23ητ2B3 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 size 12{H left (f right )= { {1} over {j ital "ωτ"} } - { {e rSup { size 8{ - jωT} } } over {j ital "ωτ"} } } {} and thus ∣ 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} } } {} The noise power o/p with white noise input N o = η 2 T τ 2 ∫ − ∞ ∞ sin π Tf π Tf 2 df = ηT 2τ 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} } } } } } {} (The integral has a value = π)

Noise Bandwidth: If a real filter with transfer fn Hf size 12{H left (f right )} {} centered at fo size 12{f rSub { size 8{o} } } {} is used, we can consider an equivalent rectangular filter centered at fo size 12{f rSub { size 8{o} } } {} with a bandwidth BN size 12{B rSub { size 8{N} } } {} passing the same noise power. BN 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 size 12{N rSub { size 8{o} } left ( ital "RC" right )= { {π} over {2} } ηf rSub { size 8{c} } } {} For a rectangular filter we have 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} } } {} Setting N0(RC)=N0(rect) size 12{N rSub { size 8{0} } \( ital "RC" \) =N rSub { size 8{0} } \( ital "rect" \) } {}, BN=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.