EE511 Day 7 Class Notes
Fourier Transform Continued
Laurence Hassebrook
Updated 9-12-03
Friday 9-12-03
FT continued
Parseval’s Theoem
Parseval’s theorem gives the relationship between energy in the time domain and energy in the
frequency domain. Quite often, in communications, we optimize for either maximum signal energy
and/or minimum noise energy. Parseval’s theorem allows us a mathematical way of working in
either time or frequency domain. Since, linear systems are easier to work with in the frequency
domain because it is simple multiplication, and optimization generally involves a differentiation,
parseval’s theorem allows us to mathematically represent energy in the frequency domain.
The theorem is




E   w1 t w2* t dt   W1  f W2*  f df
If w1(t)= w2(t) = w (t) then


E   wt  dt   W  f  df

2
2

where the latter integral is proportional to Power Spectral Density (PSD) and w(t)w*(t) = |w(t)|2.
Go over Table of FTs (notes dated 8-31-01)
Power Spectral Density and Auto Correlation (notes dated 8-31-01)
Deterministic definition of the PSD is obtained through average power as
Pave 
lim 1 T / 2 2
w t dt
T   T T / 2
Let the windowed version of the signal be wT(t) = w(t) rect(t/T), then average power can be written
as
Orthogonal Functions
FS
1
2