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

Systems Process Signals

System Implementation

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

Sampling x(t)

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

Sampling Rate

Sampling Sinusoidal Signals

Digital Frequency

Figure 4-3

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]