The quantization error, eq[n], in rounding is limited to the range
–Δ/2 to Δ/2, or " Δ/2 "
Δ is the quantization step.
Δ = xmax – xmin
L-1
eq[n] = xq[n] – x[n]
xmax and xmin are, respectively, the maximum and minimum amplitudes of x[n].
Aliasing is a phenomenon that occurs when the sampling frequency is lower than the Nyquist Rate.
Nyquist Rate is twice the maximum frequency component of the analog signal.
Nyquist Frequency is half the sampling frequency of a discrete signal. This is sometimes called the folding frequency of a sampling system.
Sampling and quantization of an analog signal
.
SAMPLING THEOREM
To avoid aliasing, Sampling Theorem states that the sampling frequency must be greater than or equal to twice the highest frequency present in the analog signal.
Fs 2Fmax
Allowable Range
In the coding process, the ADC assigns a unique binary number to each quantized level.
2b L or b log L/log2
bFs Bit Rate
where: b = number of bits
L= number of quantization levels
Fs= sampling frequency
Example:
Consider the signal
x[n] = 0.9n n 0
0 n < 0
the quantization step used is 0.1
Reference:
Digital Signal Processing: Principles, Algorithms, and Applications by Proakis, John G. c2000 Published by Pearson Education Asia Pte Ltd
4) The discrete time signal is quantized with a quantization step of 0.02. How many bits are required in the A/D converter?
5) An analog signal is sampled at the Nyquist rate of 8MHz and quantized into 180 levels. Find the time duration T (in seconds) of 1 bit of the binary signal.
Assignment 1:
1) Consider a continuous time signal x(t)=3sin(100πt). The signal is sampled at a rate of 600 samples per second. a) Find the DT signal after sampling b) For uniform quantization step of 0.5, find the number of bits and quantizing levels for this signal. c)Populate the table similar to the one on slide #12 using n 12. Plot.
Answer in 1a)
x =
0
2.4271
2.8532
0.9271
-1.7634
-3.0000
-1.7634
0.9271
2.8532
2.4271
0.0000
Example:
1) Consider the analog signal x(t)=3sin(100πt)
a) Sketch the signal x(t) for 0 t 30ms
(use 0 and multiple of 3ms). Do your
calculations up to 3 decimal places.
b) The signal x(t) is sampled at
300 samples/sec, determine the
normalized frequency of the discrete
time signal x[n].
c) At what value of t is x(t) at its maximum?
minimum?
Continuation of Assignment 1:
2) Determine the bit rate and the resolution in the sampling of analog signal with x[n] of 1.02 Vpp, sampling rate of 20 samples per second and 8-bit ADC is used.
3) An analog television signal containing frequencies up to 4.5MHz is to be coded by an ADC. The sampling frequency is 20% above the Nyquist rate. If there are 1024 quantizing levels, determine the transmission bit rate.
4) An analog signal is to be coded by an ADC. The signal contains significant frequencies up to 6MHz. The quantization error in any one sample value must be within ± 0.5% of the peak to peak amplitude range of the ADC. How many binary digits must each sample contain?
First 10 samples
SAMPLING
Generally, the analog signal is described by the function
Getting samples of the analog signal will result to x[n], a discrete time signal, generally represented as
Example:
2) Consider the analog signal (t is in seconds) x(t)=3cos(50πt)+10sin(300πt)-cos(100πt)
a)What are the frequencies present in this signal? b) What must be the sampling frequency to avoid aliasing?
3) Consider the analog signal x(t)=3cos(100πt). a)If the signal is sampled at 100Hz, what is the discrete time signal after sampling? b)If Fs=75Hz, what is x[n] after sampling?Compare your answer with (a).
Analog to Digital Conversion
Sampling is the process of converting a signal from continuous time to discrete time, obtained by taking samples of the analog signal.
Quantization is the conversion of a discrete-time continuous-valued signal into a discrete-time discrete-valued signal.
Coding is the process wherein each discrete value of the quantized signal is represented by a b-bit binary sequence.
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