Analog-to-Digital Conversion

• The presentation covers analog-to-digital conversion.
• It is presented by Engr. Julian Clement C. Villanueva, Instructor at Mapúa University.

Introduction

• Most signals of practical interest, like speech, biological, seismic, radar, sonar, audio, and video signals are analog.
• Analog signals must be converted into digital form for digital processing.
• Analog-to-digital (A/D) conversion is the procedure for converting analog signals for digital procesing, and A/D converters (ADCs) are the corresponding devices.

ADC Process

• Conceptually, A/D conversion is a three-step process:
• Sampling: Converts the analog signal into a discrete-time signal.
• Quantization: Converts the discrete-time signal into a quantized signal.
• Coding: Converts the quantized signal into a digital signal.

Sampling

• Sampling converts a continuous-time signal into a discrete-time signal by taking "samples" at discrete-time instances.
• Periodic or uniform sampling is commonly used.
• The relation x(n) = xa(nT) describes what type of sampling to use.
• x(n) is the discrete-time signal.
• xa(t) is the analog signal.
• T is every second
• The sampling period (or sample interval) is the time interval T between successive samples.
• The sampling rate (or sampling frequency), Fs, is the reciprocal of the sampling period (1/T) and is measured in samples per second or hertz.
• Formulas for sampling include
• t = nT = n/Fs
• xa(t) = A cos(2πFt + θ)
• xa(nT) = A cos(2πF(n/Fs) + θ)
• x(n) = A cos(2πn(f) + θ)
• f = F/Fs

Normalized Frequency

• Normalized frequency, f = F/Fs
• Angular frequency, ω = Ω/Fs = ΩT
• F = CT frequency (cycles/second or Hertz)
• Fs = sampling frequency (samples/second)
• f = DT relative or normalized frequency
• ω = DT angular frequency (radians/second)
• Ω = CT angular frequency (radians/second)
• F and Ω ranges for CT sine signals: -∞ ≤ F ≤ ∞ and -∞ ≤ Ω ≤ ∞
• Ranges for discrete-time sinusoids: -π ≤ ω ≤ π and -1/2 ≤ f ≤ 1/2
• Highest frequency in a discrete-time signal: ω = π or f = 1/2 Fmax = Fs/2
• Ωmax = π/T

Examples of Applications of Frequency Relation:

• Considering two analog sinusoidal signals sampled at Fs = 40 Hz
• x1(t) = cos(2π(10)t) and x2(t) = cos(2π(50)t).
• The corresponding discrete-time sequences are:
• x1(n) = cos(2π(10/40)n) = cos(πn/2).
• x2(n) = cos(2π(50/40)n) = cos(5πn/2).
• Discrete-time sinusoids whose frequencies are separated by an integer multiple of 2π are identical: cos(5πn/2) = cos(πn/2)

Further Conclusions

• An infinite number of continuous-time sinusoids can be represented by sampling the same discrete-time signal.
• Given a sequence x(n), there is ambiguity as to which continuous-time signal xa(t) these values represent.
• Frequencies Fk = F0 ± kFs, where -∞ < k < ∞ (k integer), are indistinguishable from the frequency F0 after sampling and are aliases of F0. 1
• F2 = (1/8) Hz
• F1 = (−7/8) Hz
• F1 is an alias of F2 because 1/8= (-7/8) +(1)(1)

Sampling Theorem

• The sampling rate required to convert analog signals to digital signals can be specified if the maximum frequency content of the general class of signals is known.
• For speech signals, this is up to 3kHz.
• For TV signals, this is up to 5MHz
• F ≤ Fs/2
• Fs ≥ 2F
• If the highest frequency in an analog signal xa(t) is Fmax (bandwidth B and the signal is sampled at rate Fs ≥ 2Fmax, then analog signal xa(t) can be exactly recovered from its sample values.
• The minimum sampling frequency (Fs = 2Fmax) ,is also called the Nyquist frequency or FN.
• FN = 2Fmax
• In an example where xa(t) = 3 cos 50πt + 10 sin 300πt - cos 100πt
• The frequency components are 25Hz, 150Hz, and 50Hz Fmax = 150Hz
• The Nyquist Frequency is FN = 2(150Hz) = 300Hz.

Example 2

• xa(t) = 3 cos 2000πt + 5 sin 6000πt + 10 cos 12,000πt
• The Nyquist rate is FN = 2Fmax = 2(6kHz) = 12kHz.
• If the signal is sampled at Fs = 5000 samples/s x(n) = 3 cos [2πn(1000/5000)] + 5 sin [2πn(3000/5000)] + 10 cos [2πn(6000/5000)].
• Simplify to get:
• x(n) = 3 cos(2π/5) n+5 sin(6π/5) n+10 cos((12π/5)n. x(n)3 cos(2π/5)n10 cos(2π3+5 sin(4π/5)n+10 cos(4π/5).
• It can also be expressed as:
• x(n) = 13 cos[2π(1/5)n] - 5 sin[2π(2/5)n].
• Therefore, y(t) = 13 cos[2π(1000)t] – 5 sin[2π(2000)t].

Quantization

• Quantization converts a discrete-time continuous-amplitude signal into a digital signal by expressing each sample value as a finite number of digits.
• Quantization error (or noise) is the error introduced in representing the continuous-valued signal by a finite set of discrete value levels.
• Sampled signal: 2.435493726482947273...
• Quantized signal: 2.4355 (rounded-off) or Q[x(n)].
• Quantization error sequence: eq(n) = xq(n) – x(n).

Quantization Process

• discrete-time sampled signal xa(t) = 0.9t with t ≥ 0. sampling frequency Fs = 1Hz
• The levels for Xq(n) are determined using Truncation and Rounding
• The quantization error eq(n) is the difference between the Discrete-time signal and the Quantized Signal
• The quantization error eq(n) in rounding is limited to the range Δ/2 ≤ eq(n) ≤ Δ/2
• The instantaneous quantization error cannot exceed half of the quantization step Δ = (Хmax — Хmin)/(L-1) if L is the number of quantization levels.
• An increase in the number of quantization levels, L, results in a reduced quantization step size, improving the accuracy of the quantizer and reducing this error.

Quantization Quality and Power

• The quality of the output of the A/D converter is measured by the signal-to-quantization noise ratio (SQNR).
• SQNR provides the ratio of the signal power to the noise power.
• Quantization mean-square error power: Pq = A2/3 /22b
• Average power of a signal: Px = A2/2
• A is the amplitude of the CT signal.
• b is the number of bits used.
• SQNR = (Px/Pq) = (3/2) * 22b
• In decibels, SQNR(dB) = 10 log SQNR ≈ 1.76 + 6.02b

Coding

• The coding process in an A/D converter assigns a unique binary number to each quantization level.
• With a word length of b bits, 2b binary numbers can be created
• The relation is 2b ≥ L or b ≥ log2 L.
• If there are 11 quantization levels from 0 to 1, the minimum number of bits for coding is 4, since it is the smallest number : b ≥ log211 & 4 ≥ log211.

Digital-to-Analog-Conversion

• A digital-to-analog (D/A) converter is used to convert a digital signal into an analog signal by interpolating between samples.
• The sampling theorem specifies the optimum interpolation for a band limited signal, but is too complicated.