Modulation and Sampling

1
Modulation and Sampling

• Analog and Digital Communications
• Autumn 2005-2006

2
The Dirac Delta Function

• A generalized function

3
FT of 1 and exponentials
Fourier transform of RF Pulse
4
Modulation Revisited
Multiplication in time domain is the same as
convolution in the frequency domain
5
Other Functions
Signum Function
Unit step Function
6
FT of Periodic Signals
7
FT of Periodic Signals
8
Sampling

• First consider modulation
• Product with Cosine in time domain
• Convolution with two impulses in frequency domain
• Next consider sampling
• Product with a train of impulses in time domain
• Convolution with a train of impulses in the
  frequency domain
• Nyquist sampling theorem
• A bandlimited signal -B, B can be
  characterized by its samples taken every 1/(2B)
  seconds. i.e., 2B samples per second
• Undersampling leads to aliasing