Power Spectral Density Saravanan Vijayakumaran sarva@ee.iitb.ac.in - PowerPoint PPT Presentation

Power Spectral Density Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay April 10, 2013 1 / 9

Power Spectral Density • Fourier transform � ∞ X ( f ) = x ( t ) exp ( − j 2 π ft ) dt −∞ X ( f ) = F ( x ( t )) • Inverse Fourier transform � ∞ x ( t ) = X ( f ) exp ( j 2 π ft ) df −∞ F − 1 ( X ( f )) x ( t ) = Definition (Power Spectral Density of a WSS Process) The power spectral density of a wide-sense stationary random process is the Fourier transform of the autocorrelation function. S X ( f ) = F ( R X ( τ )) 2 / 9

Motivating the Definition of Power Spectral Density X ( t ) LTI System Y ( t ) • Consider an LTI system with impulse response h ( t ) which has random processes X ( t ) and Y ( t ) as input and output � ∞ Y ( t ) = h ( τ ) X ( t − τ ) d τ −∞ • If X ( t ) is a wide-sense stationary random process, then Y ( t ) is also wide-sense stationary with autocorrelation function � ∞ � ∞ R Y ( τ ) = h ( τ 1 ) h ( τ 2 ) R X ( τ − τ 1 + τ 2 ) d τ 1 d τ 2 −∞ −∞ • Setting τ = 0, we can express the average power in the output process as � ∞ � ∞ � � Y 2 ( t ) R Y ( 0 ) = E = h ( τ 1 ) h ( τ 2 ) R X ( τ 2 − τ 1 ) d τ 1 d τ 2 −∞ −∞ 3 / 9

Motivating the Definition of Power Spectral Density • Let H ( f ) be the Fourier transform of the impulse response h ( t ) � ∞ h ( τ 1 ) = H ( f ) exp ( j 2 π f τ 1 ) df −∞ • Substituting the above equation into the average power equation we get � ∞ � ∞ � � Y 2 ( t ) E = h ( τ 1 ) h ( τ 2 ) R X ( τ 2 − τ 1 ) d τ 1 d τ 2 −∞ −∞ � ∞ � ∞ �� ∞ � H ( f ) e j 2 π f τ 1 df = h ( τ 2 ) R X ( τ 2 − τ 1 ) d τ 1 d τ 2 −∞ −∞ −∞ � ∞ � ∞ �� ∞ � R X ( τ ) e − j 2 π f τ d τ df h ( τ 2 ) e j 2 π f τ 2 d τ 2 = H ( f ) −∞ −∞ −∞ � ∞ � ∞ H ( f ) H ∗ ( f ) R X ( τ ) e − j 2 π f τ d τ df = −∞ −∞ � ∞ � ∞ R X ( τ ) e − j 2 π f τ d τ df | H ( f ) | 2 = −∞ −∞ � ∞ | H ( f ) | 2 S X ( f ) df = −∞ 4 / 9

Motivating the Definition of Power Spectral Density • The output power and power spectral density are related by � ∞ � � Y 2 ( t ) | H ( f ) | 2 S X ( f ) df E = −∞ • Let the LTI system be an ideal narrowband filter with magnitude response given by � 1 | f ± f c | ≤ ∆ f | H ( f ) | = 2 | f ± f c | > ∆ f 0 2 1 | H ( f ) | − f c f c � � Y 2 ( t ) E ≈ ( 2 ∆ f ) S X ( f c ) 5 / 9

Properties of Power Spectral Density • The power spectral density and autocorrelation function form a Fourier transform pair � ∞ S X ( f ) = R X ( τ ) exp ( − i 2 π f τ ) d τ −∞ � ∞ R X ( τ ) = S X ( f ) exp ( i 2 π f τ ) df −∞ • Power spectral density is a non-negative and even function of f • Zero-frequency PSD value equals area under autocorrelation function � ∞ S X ( 0 ) = R X ( τ ) d τ −∞ • Power of X ( t ) equals area under power spectral density � ∞ � � X 2 ( t ) E = S X ( f ) df −∞ • If X ( t ) is passed through an LTI system with frequency response H ( f ) to get Y ( t ) S Y ( f ) = | H ( f ) | 2 S X ( f ) 6 / 9

White Noise • A wide-sense stationary random process with flat power spectral density S W ( f ) = N 0 2 where N 0 has dimensions Watts per Hertz • White noise has infinite power and is not physically realizable • Models a situation where the noise bandwidth is much larger than the signal bandwidth • The corresponding autocorrelation function is given by R N ( τ ) = N 0 2 δ ( τ ) where δ is the Dirac delta function 7 / 9

Reference • Chapter 1, Communication Systems , Simon Haykin, Fourth Edition, Wiley-India, 2001. 8 / 9

Questions? 9 / 9