Quantization in Communications Systems II: Uniform Quantizers and Quantization Noise, Lecture notes of Mathematics

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ECE 442

Communications Systems II

ECE 442

Lecture 3: Quantization

Mohamed Ibrahim

ECE 442

From Continuous to Digital

r In continuous wave (CW) modulation, some parameter of a carrier wave is varied according to the message signal r In pulse modulation, some parameter of a pulse train is varied in accordance with the message signal Þ The pulses are then transmitted over the physical medium r Two families of pulse modulation Þ Analog pulse modulation

• Analog form at discrete times Þ Digital pulse modulation
• Discrete in both time and amplitude
• Sampling in time and Quantization of amplitude!

ECE 442

r Quantizers can be Þ Uniform where the representation levels are equally spaced Þ Non-uniform (considered later) r Uniform Quantizers Þ Mid-tread: origin lies in the middle of a tread Þ Mid-rise: origin lies in the middle of the rising part of the staircase graph Þ Both are symmetric around the origin

Uniform Quantizers

ECE 442

r The error caused by the difference between the input signal! and the output signal " is referred to as quantization noise

Quantization Noise

ECE 442

r Since the mean of! is zero, the variance is the same as the mean square value r Using the PDF of! r Let " denote the number of bits per sample used in the binary code Þ # = 2 & → " = log+# and therefore Þ Thus the variance can be written as

Quantization Noise

ECE 442

r If P denotes the average power of the message signal !(#) we can find the output signal-to-noise ratio as r The output %&' of the quantizer increases exponentially with increasing the number of bits per sample ' Þ To increase ' we need to increase () r So, using a binary code for the representation of a message signal provides a more efficient method for the trade-off of increased bandwidth for improved noise performance than either FM or PPM Þ FM and PPM are limited by receiver noise, while quantization is limited by quantization noise

Output Signal to Noise Ratio

ECE 442

r The process of COMpressing and exPANDING the signal is called companding Þ Companding is a technique used to reduce the number of bits required in ADC or DAC while achieving comparable SQNR r After compression, the signal can be input to a uniform quantizer Þ At the receiver, the signal is expanded by an inverse operation r Two standard logarithm based companding techniques Þ !-law companding Þ A-law companding

Companding

ECE 442

Communications Systems II

ECE 442

Thank You!