Lecture 2

Analog to digital conversion&

Basic discrete signals

ADC process

Anti-aliasingfilter

ADCx(t)x [n]

DSP

1. Time discretization2. 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 65356 possible values : :• N bits 2N 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 processAnalog

filterDAC 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 signalsWhy ?

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 < 0u[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]