Lecture 16: Properties of the Fourier Transform

1
Lecture 16 Properties of the Fourier Transform
2
(No Transcript)
3
(No Transcript)
4
(No Transcript)
5
(No Transcript)
6
In the time domain, this is amplitude modulation
(AM)
Baseband signal (e.g. audio).
In the time domain, modulation is complicated.
However in the frequency domain, it amounts to
nothing more than shifting in frequency.
7
(No Transcript)
8
10) Parseval relation
Integral square is preserved between time and
frequency.
9
Application of Fourier to LTI systems

• Remarks
• The Fourier transform takes convolutions into
  products.
• Similar to Laplace, but here we do not require
  causality,
• and x(t) could be nonzero for all time.

10
(No Transcript)
11
Interpretation system response to a sinusoid
12
Application amplitude demodulation
y(t)
Demodulator

x(t)
v(t)
Q What operation can we do to make the received
signal v(t) equal to the message x(t)?
13
Solution another modulation lowpass filtering

14
Fourier and Convolution