the power spectral density at the filter output is white within the bandwidth B

The power spectral density at the filter output is

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the power spectral density at the filter output is white within the bandwidth B N and zero elsewhere, forming an equivalent rectangular spectral density; (b) the area under this rectangular spectral density is equal to the area under the spectral density at the filter output.
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ERG2310A-I p. I-77 Equivalent Noise Bandwidth Let ω 0 be the midband frequency of the system , . ) ( ) ( 2 2 1 ) ( 2 0 2 2 2 0 2 N B B o B H d H t n N N ω η ω ω η π π π = = 2 0 0 2 ) ( ) ( 2 1 ω ω ω π H d H B N = = = 0 2 2 2 ) ( 2 ) ( 2 2 1 ) ( ω ω π η ω ω η π d H d H t n o Q ω o ω | Η(ω o )| 2 2πΒ Ν Η(ω ) ( ω o = 0 for low-pass filter)
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ERG2310A-I p. I-78 Equivalent Noise Bandwidth Example: Computer the –3dB bandwidth and the equivalent noise bandwidth of a filter with the following magnitude transfer characteristic: 4 1 1 ) ( ω ω + = H As | H(0) |=1, the –3dB bandwidth is found by solving 2 1 ) ( 3 = dB H ω Æ ω 3dB =1 or f 3dB =1/(2 π ) = 0.159Hz = = + = = 0 4 2 0 0 2 177 . 0 8 2 1 1 2 1 ) ( ) ( 2 1 Hz d H d H B N ω ω π ω ω ω π Using
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ERG2310A-I p. I-79 Signal-to-Noise Ratio (SNR) It is a dimensionless ratio of signal power to noise power . ) ( / ) ( / 2 2 t n t s N S = It is quite common to express the signal-to-noise ratio in decibels [ ] [ ] . ) ( / ) ( log 10 / 2 2 10 t n t s N S dB = * Both of these mean-square values assume zero mean value unless stated otherwise. SNR is commonly used to describe the quality of a signal.
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ERG2310A-I p. I-80 Noise Figure Let the input and output signal voltages (or currents) in a given system be s i ( t ), s o ( t ) respectively and the input and output noise voltages (or currents) be n i ( t ), n o ( t ). The input signal-to-noise ratio is , ) ( / ) ( ) / ( 2 2 t n t s N S i i i = The output signal-to-noise ratio is , ) ( / ) ( ) / ( 2 2 t n t s N S o o o = The noise figure F is defined to be the ratio of the input signal-to- noise ratio to the output signal-to-noise ratio: . ) / ( ) / ( o i N S N S F = (S/N) o h ( t ) (S/N) i Noise figure (expressed in decibel, i.e. 10log( F )) is commonly used to describe the noise performance of a system. For ideal system (no noise) Î F =1
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