Lecture 2 Analog to digital conversion & Basic discrete signals

ADC process Anti-aliasing filter ADC x(t) x [n] DSP 1.Time discretization 2.Amplitude discretization

Time discretization Shannon Sampling Theorem: The sampling frequency should be at least twice the maximum frequency of the signal. fmax < fs / 2 = 1/2T (fs/2: Nyquist frequency) Aliasing: spurious low frequencies introduced by low sampling. The first stage in ADC is an anti-aliasing low pass filter! T

1 bit  2 possible values 2 bits  4 possible values 8 bits  256 possible values 16 bits  possible values : : N bits  2 N possible values Amplitude discretization } Quantization noise

Dynamic range = (max possible value – min possible value) –If too low Good resolution Risk of saturation –If too high Poor resolution No saturation

DAC process Analog filter DAC x(t) x [n] DSP Sample and hold

Signal types Continuous time Continuous amplitude Discrete time Continuous amplitude Continuous time Discrete amplitude Digital signals

Basic digital signals Why ? Complex signals can usually be expressed as summation of simple ones. For linear DSPs, if we know the response to basic signals we can predict the response to more complex ones. They can be used as test signals for studying properties of DSPs.

Unit impulse function 1 n=0 n  [n]  [n] = 0 n ≠ 0  [n] = 1 n = 0 Unit step function n=0 n  [n] u[n] = 0 n < 0 u[n] = 1 n ≥ 0 1

Exponential function n=0 n x [n] x[n] = A  n 0 <  < 1 Sinusoidal function n=0 n x [n] x[n] = A cos(  n +  ) 1

Periodicity A signal is periodic if repeats after T values: x [n] = x [n+T] = x [n+2T] = … T is the period of the signal Exercise: Calculate the period of: a)x [n] = cos (Πn/4) b)x [n] = cos (3Πn/4)

Exercise: Draw the following sequences: 1.x [n] = u[n-2] 2.x [n] = n u[n] 3.x [n] = -3  [n+4] 4.x [n] =  n  5.x [n] = -2 u[-n-2] 6.x [n] = u[n+2] – u[n-6]

Exercise: Find the mathematical expressions of the following sequences: a)c)b)

Exercise: Given the sequence of the figure, draw the following sequences: a) x[n-2] b) x[3-n] c) x[n-1] u[n] d) x[n-1]  [n] e) x[1-n]  [n-2]