1. Today's lecture System Implementation Discrete Time signals generation
Sampling Theorem
Sampling Rate
Sinusoidal Signal Sampling
Concept of Aliasing
Spectrum for Discrete Time Domain
Digital Frequency

2. Systems Process Signals

3. System Implementation

4. Discrete-time Signals
There are two ways to obtain discrete time signal.
a) Can be computed using a formula such as
x[n]= n2 - 5n +3
b) Sample the continous time signal.
x(t) = A cos (ωot + )
x[n] = x(nTs)
Signal represented as an indexed sequence of numbers which are samples of x(t) at Ts interval

5. Sampling x(t)

6. Sampling Theorem
Continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x(k Ts) if samples taken at rate fs > 2 fmax
Nyquist rate = 2 fmax
Nyquist frequency = fs / 2
Sampling theorem also suggests that there should be two samples per cycle.
Example: Sampling audio signals
Normal human hearing is from about 20 Hz to 20 kHz

7. Sampling Rate

8. Sampling Sinusoidal Signals

9. Digital Frequency

10. Figure 4-3

11. The Concept of Aliasing
Two different cosine signals can be drawn through the same samples
x1[n] = cos(0.4πn)
x2[n] = cos(2.4πn)
x2[n] = cos(2πn + 0.4πn)
x2[n] = cos(0.4πn)
x2[n] = x1[n]