2ndyr dig comm lect1 day1
TRANSCRIPT
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Analog signal processing Vs Digital Signal Processing
In analog processing system the information is represented as an analog quantity. ie; something
that varies continuously with time. This "something" can be voltage , current etc.. . for example, the
signal that comes out of a microphone is analog in nature. The varying voltage from the microphone
is processed using an analog signal processing system known as "amplifier" , and reproduced
through a speaker.
Where in Digital Signal Processing system , the information is represented in digital format.
The signals that are to be processed is converted into numerical form before any processing. This
conversion is known as sampling. The information contained in an analog signal is first converted to
digital samples which are equally spaced in time. The figure below shows an analog signal and its
sampled version.
Each of these samples are converted in to a numerical value and stored in the computer's memory.
Processing is then done on these samples. The sampling in a Digital Signal Processing system is
governed by a theorem known as sampling theorem.
Sampling theorem
According to sampling theorem, an analog signal can be exactly reconstructed from its samples
if the sampling rate is at least twice the highest frequency component present in the signal. This
means, if the signal contains the highest frequency component of 1KHz, the sampling rate must be
at least 2 Kilo Samples/ Second. This sampling rate is also known as Nyquist rate. What if
sampling theorem is not satisfied..? violation of sampling theorem means , sampling the signal at a
sample rate less than Nyquist rate. This leads to unacceptable distortion in the signal . this distortion
is known as aliasing. Due to aliasing , the components present in the input signal whose frequency
is higher Nyquist frequency will be sampled as a signal of lower frequency. And this aliased
samples in turn corrupts the original signal which is lower than Nyquist frequency. Due to the
presence of aliasing , a signal that contains frequency components higher than Nyquist frequencycan not be reconstructed from it's samples. The effect of aliasing is shown in the figure below.
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The signal shown in black color is the original signal. While the signal shown in violate color is the
aliased signal because of improper sampling. From the figure it is obvious that the aliased signal
will be present in the sampled data as a lower frequency signal. And this will affect the original
content at that frequency.
It should be noted that the information in between two consecutive samples is lost forever. But
still the entire signal can be reconstructed from the samples as long as Sampling theorem is
satisfied.
Sampling and quantization
Sampling can be viewed theoretically as multiplying the original signal with a train of pulses
with unit amplitude. This leaves the original signal with information at descrete points with
magnitude of signal present in the original signal at that position. This is illustrated in the figurebelow.
This can be implemented using an AND gate as shown below.
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Now samples of the input signal is taken at discrete points. But in order to manipulate the signal
with a microprocessor or micro controller, these pulses has to be converted in to numbers. Before
this conversion, the pulses are quantized to some finite number of quantization levels. For example,
if the quantization levels are 0,1,2,3, etc... , a pulse with magnitude between 1 and 2 will be
quantized to either 1 or 2. This in fact introduces a noise in the sampled signal. Such noise in known
as quantization noise.
Quantization (signal processing)
Discrete signal
Digital signal (after quantization has occurred)
In digital signal processing, quantization is the process of approximating a continuous range of
values (or a very large set of possible discrete values) by a relatively small set of discrete symbols orinteger values. More specifically, a signal can be multi dimensional and quantization need not be
applied to all dimensions. Discrete signals (a common mathematical model) need not be quantized,
which can be a point of confusion.
A common use of quantization is in the conversion of a discrete signal (a sampledcontinuous
signal) into a digital signal by quantizing. Both of these steps (sampling and quantizing) are
performed in analog to digital converters with the quantization level specified inbits. A specific
example would be compact disc (CD) audio which is sampled at 44,100Hz and quantized with 16
bits (2 bytes) which can be one of 65,536 (i.e. 216) possible values per sample.
In electronics, adaptive quantization is a quantization process that varies the step size based on thechanges of the input signal, as a means of efficient compression.Two approaches commonly used
are forward adaptive quantization and backward adaptive quantization.
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Mathematical description
Quantization is referred to as scalar quantization, since it operates on scalar (as opposed to multi
dimensionalvector) input data. In general, a scalar quantization operator can be represented as
whereng an integer result that is sometimes referred to as the quantization index,
f(x) and g(i) are arbitrary real valued functions.
The integer valued quantization index i is the representation that is typically stored or transmitted,
and then the final interpretation is constructed using g(i) when the data is later interpreted.
In computer audio and most other applications, a method known as uniform quantization is the most
common. There are two common variations of uniform quantization, called mid rise and mid tread
uniform quantizers.
Ifx is a real valued number between 1 and 1, a mid rise uniform quantization operator that uses Mbits of precision to represent each quantization index can be expressed as
.
In this case thef(x) and g(i) operators are just multiplying scale factors (one multiplier being the
inverse of the other) along with an offset in g(i) function to place the representation value in the
middle of the input region for each quantization index. The value 2 (M 1) is often referred to as
the quantization step size. Using this quantization law and assuming thatquantization noise is
approximately uniformly distributed over the quantization step size (an assumption typicallyaccurate for rapidly varyingx or highM) and further assuming that the input signalx to be
quantized is approximately uniformly distributed over the entire interval from 1 to 1, the signal to
noise ratio (SNR) of the quantization can be computed as
.
From this equation, it is often said that the SNR is approximately 6 dB per bit.
For mid tread uniform quantization, the offset of 0.5 would be added within the floor function
instead of outside of it.
Sometimes, mid rise quantization is used without adding the offset of 0.5. This reduces the signal to
noise ratio by approximately 6.02 dB, but may be acceptable for the sake of simplicity when the step
size is small.
In digital telephony, two popular quantization schemes are the 'A law ' (dominant in Europe) and '
law' (dominant inNorth AmericaandJapan). These schemes map discrete analog values to an 8 bit
scale that is nearly linear for small values and then increases logarithmically as amplitude grows.
Because the human ear's perception ofloudness is roughly logarithmic, this provides a higher signal
to noise ratio over the range of audible sound intensities for a given number of bits.
http://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Quantization_noisehttp://en.wikipedia.org/wiki/Quantization_noisehttp://en.wikipedia.org/wiki/Uniform_distribution_(continuous)http://en.wikipedia.org/wiki/Signal_to_noise_ratiohttp://en.wikipedia.org/wiki/Signal_to_noise_ratiohttp://en.wikipedia.org/wiki/Signal_to_noise_ratiohttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Digital_telephonyhttp://en.wikipedia.org/wiki/A-law_algorithmhttp://en.wikipedia.org/wiki/A-law_algorithmhttp://en.wikipedia.org/wiki/Europehttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/North_Americahttp://en.wikipedia.org/wiki/North_Americahttp://en.wikipedia.org/wiki/North_Americahttp://en.wikipedia.org/wiki/Japanhttp://en.wikipedia.org/wiki/Japanhttp://en.wikipedia.org/wiki/Loudnesshttp://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Quantization_noisehttp://en.wikipedia.org/wiki/Uniform_distribution_(continuous)http://en.wikipedia.org/wiki/Signal_to_noise_ratiohttp://en.wikipedia.org/wiki/Signal_to_noise_ratiohttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Digital_telephonyhttp://en.wikipedia.org/wiki/A-law_algorithmhttp://en.wikipedia.org/wiki/Europehttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/Mu-law_algorithmhttp://en.wikipedia.org/wiki/North_Americahttp://en.wikipedia.org/wiki/Japanhttp://en.wikipedia.org/wiki/Loudness -
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NyquistShannon sampling theorem
Fig: Hypothetical spectrum of a bandlimited signal as a function of frequency
The NyquistShannon sampling theorem is a fundamental result in the field ofinformation
theory, in particulartelecommunications andsignal processing.Samplingis the process of
converting a signal (for example, a function of continuous time or space) into a numeric sequence (a
function of discrete time or space). The theorem states:[1]
If a functionx(t) contains no frequencies higher thanBcps, it is completely determined
by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
In essence the theorem shows that an analog signal that has been sampled can be perfectly
reconstructed from the samples if the sampling rate exceeds 2B samples per second, whereB is the
highest frequency in the original signal. If a signal contains a component at exactlyB hertz, then
samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon's
statement notwithstanding.
More recent statements of the theorem are sometimes careful to exclude the equality condition; that
is, the condition is ifx(t) contains no frequencies higher than or equal toB; this condition is
equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly
frequencyB.
The assumptions necessary to prove the theorem form a mathematical model that is only an
idealization of any real world situation. The conclusion that perfect reconstruction is possible is
mathematically correct for the model but only an approximation for actual signals and actualsampling techniques.
The theorem also leads to a formula for reconstruction of the original signal. The constructive proof
of the theorem leads to an understanding of the aliasing that can occur when a sampling system does
not satisfy the conditions of the theorem.
The NyquistShannon sampling theorem is also known to be a sufficient condition. The field of
Compressed sensing provides a stricter sampling condition when the underlying signal is known to
be sparse. Compressed sensing specifically yields a sub Nyquist sampling criterion.
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Downsampling
When a signal isdownsampled, the sampling theorem can be invoked via the artifice of resampling
a hypothetical continuous time reconstruction. The Nyquist criterion must still be satisfied withrespect to the new lower sampling frequency in order to avoid aliasing. To meet the requirements of
the theorem, the signal must usually pass through a low pass filter of appropriate cutoff frequency
as part of the downsampling operation. This low pass filter, which prevents aliasing, is called an
anti aliasing filter .
Critical frequency
Fig: A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1
and 1. That is, they all are aliases of each other, even though their frequency is not above half the sample
rate.
The Nyquist rate is defined as twice the bandwidth of the continuous time signal. The sampling
frequency must be strictly greaterthan the Nyquist rate of the signal to achieve unambiguous
representation of the signal. This constraint is equivalent to requiring that the system's Nyquist
frequency (also known as critical frequency, and equal to half the sample rate) be strictly greater
than the bandwidth of the signal. If the signal contains a frequency component at precisely the
Nyquist frequency then the corresponding component of the sample values cannot have sufficient
information to reconstruct the Nyquist frequency component in the continuous time signal because
of phase ambiguity. In such a case, there would be an infinite number of possible and different
sinusoids (of varying amplitude and phase) of the Nyquist frequency component that are
represented by the discrete samples.
As an example, consider this family of signals at the critical frequency:
Where the samples
are in every case just alternating 1 and +1, for any phase . There is no way to determine either the
amplitude or the phase of the continuous time sinusoid x(t) thatx(nT) was sampled from. This
ambiguity is the reason for the strictinequality of the sampling theorem's condition.
http://en.wikipedia.org/wiki/Downsamplinghttp://en.wikipedia.org/wiki/Downsamplinghttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Anti-aliasing_filterhttp://en.wikipedia.org/wiki/Nyquist_ratehttp://en.wikipedia.org/wiki/Continuous-timehttp://en.wikipedia.org/wiki/Continuous-timehttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svghttp://en.wikipedia.org/wiki/Downsamplinghttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Anti-aliasing_filterhttp://en.wikipedia.org/wiki/Nyquist_ratehttp://en.wikipedia.org/wiki/Continuous-timehttp://en.wikipedia.org/wiki/Nyquist_frequencyhttp://en.wikipedia.org/wiki/Nyquist_frequency -
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Analog to digital converterAn analog to digital converter (abbreviated ADC, A/D or A to D) is a device which converts
continuoussignals to discretedigital numbers. The reverse operation is performed by a digital to
analog converter (DAC).
Typically, an ADC is anelectronic device that converts an input analog voltage (or current) to a
digital number. However, some non electronic or only partially electronic devices, such as rotary
encoders, can also be considered ADCs. The digital output may use different coding schemes, such
as binary, Gray code or two's complement binary.
Concepts
Resolution
The resolution of the converter indicates the number of discrete values it can produce over the rangeof analog values. The values are usually stored electronically in binaryform, so the resolution is
usually expressed in bits. In consequence, the number of discrete values available, or "levels", is
usually a power of two. For example, an ADC with a resolution of 8 bits can encode an analog input
to one in 256 different levels, since 28 = 256. The values can represent the ranges from 0 to 255 (i.e.
unsigned integer) or from 128 to 127 (i.e. signed integer), depending on the application.
Resolution can also be defined electrically, and expressed in volts. The voltage resolution of an
ADC is equal to its overall voltage measurement range divided by the number of discrete intervals
as in the formula:
Where:
Q is resolution in volts per step (volts per output code),
EFSR
is the full scale voltage range = VRefHi
VRefLo
,
M is the ADC's resolution in bits, and
N is the number of intervals, given by the number of available levels (output codes), which is:
N= 2M
Some examples may help:
Example 1
Full scale measurement range = 0 to 10 volts
ADC resolution is 12 bits: 212 = 4096 quantization levels (codes)
ADC voltage resolution is: (10V 0V) / 4096 codes = 10V / 4096 codes
0.00244 volts/code 2.44 mV/code
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Example 2
Full scale measurement range = 10 to +10 volts
ADC resolution is 14 bits: 214 = 16384 quantization levels (codes)
ADC voltage resolution is: (10V ( 10V)) / 16384 codes = 20V / 16384 codes
0.00122 volts/code 1.22 mV/code
Example 3
Full scale measurement range = 0 to 8 volts
ADC resolution is 3 bits: 23 = 8 quantization levels (codes)
ADC voltage resolution is: (8 V 0 V)/8 codes = 8 V/8 codes = 1 volts/code = 1000
mV/code
In practice, the smallest output code ("0" in an unsigned system) represents a voltage range which is
0.5X of the ADC voltage resolution (Q)(meaning half wide of the ADC voltage Q ) while the
largest output code represents a voltage range which is 1.5X of the ADC voltage resolution
(meaning 50% wider than the ADC voltage resolution). The otherN 2 codes are all equal in width
and represent the ADC voltage resolution (Q) calculated above. Doing this centers the code on aninput voltage that represents theMth division of the input voltage range. For example, in Example 3,
with the 3 bit ADC spanning an 8 V range, each of the Ndivisions would represent 1 V, except the
1st ("0" code) which is 0.5 V wide, and the last ("7" code) which is 1.5 V wide. Doing this the "1"
code spans a voltage range from 0.5 to 1.5 V, the "2" code spans a voltage range from 1.5 to 2.5 V,
etc. Thus, if the input signal is at 3/8ths of the full scale voltage, then the ADC outputs the "3" code,
and will do so as long as the voltage stays within the range of 2.5/8ths and 3.5/8ths. This practice is
called "Mid Tread" operation. This type of ADC can be modeled mathematically as:
The exception to this convention seems to be the Microchip PIC processor, where all Msteps are
equal width. This practice is called "Mid Rise with Offset" operation.
In practice, the useful resolution of a converter is limited by the best signal to noise ratio that can be
achieved for a digitized signal. An ADC can resolve a signal to only a certain number of bits of
resolution, called the "effective number of bits" (ENOB). One effective bit of resolution changes the
signal to noise ratio of the digitized signal by 6 dB, if the resolution is limited by the ADC. If a
preamplifier has been used prior to A/D conversion, the noise introduced by the amplifier can be animportant contributing factor towards the overall SNR.
Response type
Linear ADCs
Most ADCs are of a type known as linear, although analog to digital conversion is an inherently
non linear process (since the mapping of a continuous space to a discrete space is a piecewise constant and therefore non linear operation). The term linearas used here means that the range of
the input values that map to each output value has a linear relationship with the output value, i.e.,
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that the output value kis used for the range of input values from
m(k+ b)
to
m(k+ 1 + b),
where m and b are constants. Here b is typically 0 or 0.5. When b = 0, the ADC is referred to as
mid rise , and when b = 0.5 it is referred to as mid tread .
Non linear ADCs
If theprobability density function of a signal being digitized isuniform, then the signal to noise
ratio relative to the quantization noise is the best possible. Because this is often not the case, it's
usual to pass the signal through its cumulative distribution function (CDF) before the quantization.
This is good because the regions that are more important get quantized with a better resolution. In
the dequantization process, the inverse CDF is needed.
This is the same principle behind the companders used in some tape recorders and other
communication systems, and is related to entropymaximization. (Never confuse companders with
compressors!)
For example, a voice signal has a Laplacian distribution. This means that the region around the
lowest levels, near 0, carries more information than the regions with higher amplitudes. Because of
this, logarithmic ADCs are very common in voice communication systems to increase the dynamic
range of the representable values while retaining fine granular fidelity in the low amplitude region.
An eight bit a law or the law logarithmic ADC covers the widedynamic rangeand has a high
resolution in the critical low amplitude region, that would otherwise require a 12 bit linear ADC.
Accuracy
An ADC has several sources of errors. Quantization error and (assuming the ADC is intended to be
linear) non linearityis intrinsic to any analog to digital conversion. There is also a so called
aperture errorwhich is due to a clockjitterand is revealed when digitizing a time variant signal
(not a constant value).
These errors are measured in a unit called theLSB, which is an abbreviation for least significant bit.
In the above example of an eight bit ADC, an error of one LSB is 1/256 of the full signal range, or
about 0.4%.
Quantization error
Quantization error is due to the finite resolution of the ADC, and is an unavoidable imperfection in
all types of ADC. The magnitude of the quantization error at the sampling instant is between zero
and half of one LSB.
In the general case, the original signal is much larger than one LSB. When this happens, the
quantization error is not correlated with the signal, and has auniform distribution. Its RMS value is
thestandard deviationof this distribution, given by . In the eight bitADC example, this represents 0.113% of the full signal range.
At lower levels the quantizing error becomes dependent of the input signal, resulting in distortion.
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This distortion is created after the anti aliasing filter, and if these distortions are above 1/2 the
sample rate they will alias back into the audio band. In order to make the quantizing error
independent of the input signal, noise with an amplitude of 1 quantization step is added to the
signal. This slightly reduces signal to noise ratio, but completely eliminates the distortion. It is
known as dither.
Non linearity
All ADCs suffer from non linearity errors caused by their physical imperfections, resulting in their
output to deviate from a linear function (or some other function, in the case of a deliberately non
linear ADC) of their input. These errors can sometimes be mitigated by calibration, or prevented by
testing.
Important parameters for linearity are integral non linearity (INL) anddifferential non linearity
(DNL). These non linearities reduce the dynamic range of the signals that can be digitized by the
ADC, also reducing the effective resolution of the ADC.
Digital Ramp ADC
Conversion from analog to digital form inherently involves comparator action where the value of theanalog voltage at some point in time is compared with some standard. A common way to do that is
to apply the analog voltage to one terminal of a comparator and trigger a binary counter which
drives a DAC. The output of the DAC is applied to the other terminal of the comparator. Since the
output of the DAC is increasing with the counter, it will trigger the comparator at some point when
its voltage exceeds the analog input. The transition of the comparator stops the binary counter,
which at that point holds the digital value corresponding to the analog voltage.
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Successive Approximation ADC
Illustration of 4 bit SAC with 1 volt step size (after Tocci, Digital Systems).
The successive approximation ADC is much
faster than the digital ramp ADC because it uses
digital logic to converge on the value closest to
the input voltage. A comparatorand a DAC areused in the process.
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Flash ADCIllustrated is a 3 bit flash ADC with resolution 1 volt (after Tocci). The resistor net and comparators
provide an input to the combinational logic circuit, so the conversion time is just the propagation
delay through the network it is not limited by the clock rate or some convergence sequence. It is
the fastest type of ADC available, but requires a comparator for each value of output (63 for 6 bit,
255 for 8 bit, etc.) Such ADCs are available in IC form up to 8 bit and 10 bit flash ADCs (1023
comparators) are planned. The encoder logic executes a truth table to convert the ladder of inputs to
the binary number output.
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Digital to analog converterIn electronics, a digital to analog converter (DAC or D to A ) is a device for converting a digital
(usually binary) code to an analog signal (current, voltage or electric charge).
Ananalog to digital converter (ADC) performs the reverse operation.
Basic ideal operation
Ideally sampled signal. Signal of a typical interpolating DAC output
A DAC converts an abstract finite precision number (usually a fixed point binary number) into a
concrete physical quantity (e.g., avoltage or a pressure). In particular, DACs are often used to
convert finite precision time series data to a continually varying physical signal.
A typical DAC converts the abstract numbers into a concrete sequence ofimpulses that are then
processed by a reconstruction filter uses some form ofinterpolation to fill in data between the
impulses. Other DAC methods (e.g., methods based onDelta sigma modulation ) produce a pulse
density modulated signal that can then be filtered in a similar way to produce a smoothly varying
signal.
By theNyquistShannon sampling theorem, sampled data can be reconstructed perfectly providedthat its bandwidth meets certain requirements (e.g., a baseband signal withbandwidth less than the
Nyquist frequency). However, even with an ideal reconstruction filter, digital sampling introduces
quantization error that makes perfect reconstruction practically impossible. Increasing the digital
resolution (i.e., increasing the number ofbits used in each sample) or introducing samplingdither
can reduce this error.
Practical operation
Instead of impulses, usually the sequence of numbers update the analogue voltage at uniform
sampling intervals.
These numbers are written to the DAC, typically with a clock signal that causes each number to be
latched in sequence, at which time the DAC output voltage changes rapidly from the previous value
to the value represented by the currently latched number. The effect of this is that the output voltage
is heldin time at the current value until the next input number is latched resulting in apiecewise
constant or 'staircase' shaped output. This is equivalent to azero order hold operation and has an
effect on the frequency response of the reconstructed signal.
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Piecewise constant signal typical of a zero order (non interpolating) DAC output.
The fact that practical DACs output a sequence of piecewise constant values or rectangular pulses
would cause multiple harmonics above the nyquist frequency. These are typically removed with a
low pass filter acting as a reconstruction filter.
However, this filter means that there is an inherent effect of the zero order hold on the effective
frequency response of the DAC resulting in a mild roll off of gain at the higher frequencies (often a
3.9224 dB loss at the Nyquist frequency) and depending on the filter, phase distortion. This high
frequency roll off is the output characteristic of the DAC, and is not an inherent property of the
sampled data.
Applications
Audio
Most modern audio signals are stored in digital form (for example MP3s and CDs) and in order to
be heard through speakers they must be converted into an analog signal. DACs are therefore found
in CD players,digital music players, and PC sound cards.
Specialist stand alone DACs can also be found in high end hi fi systems. These normally take the
digital output of aCD player(or dedicated transport) and convert the signal into aline level output
that can then be fed into apre amplifier stage.
Similar digital to analog converters can be found in digital speakerssuch as USB speakers, and in
sound cards.
Video
Video signals from a digital source, such as a computer, must be converted to analog form if theyare to be displayed on an analog monitor. As of 2007, analog inputs are more commonly used than
digital, but this may change as flat panel displayswith DVIand/or HDMI connections become more
widespread. A video DAC is, however, incorporated in any Digital Video Player with analog
outputs. The DAC is usually integrated with some memory (RAM), which contains conversion
tables forgamma correction, contrast and brightness, to make a device called aRAMDAC.
A device that is distantly related to the DAC is the digitally controlled potentiometer, used to control
an analog signal digitally.
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DAC types
The most common types of electronic DACs are:
Four Bit D/A ConverterOne way to achieveD/A conversion is to use asumming amplifier.
This approach is not satisfactory for a large number of bits because it requires too much
precision in the summing resistors. This problem is overcome in the R 2R network DAC.
Summing Amplifier
This is an example of an inverting amplifier
of gain=1 with multiple inputs. More than
two inputs can be used, for example in an
audio mixer circuit. The input resistors can
be unequal, giving a weighted sum.
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R 2R Ladder DAC
The summing amplifier with the R 2R ladder of
resistances shown produces the output
where the D's take the value 0 or 1. The digital inputs
could be TTL voltages which close the switcheson a
logical 1 and leave it grounded for a logical 0. This is
illustrated for 4 bits, but can be extended to any number
with just the resistance values R and 2R.
R 2R Ladder DAC Details
the Pulse Width Modulator, the simplest DAC type. A stable current or voltage is switched
into a low pass analog filter with a duration determined by the digital input code. This
technique is often used for electric motor speed control, and is now becoming common in
high fidelity audio.
Oversampling DACs or Interpolating DACs such as the Delta Sigma DAC , use a pulse
density conversion technique. Theoversampling technique allows for the use of a lower
resolution DAC internally. A simple 1 bit DAC is often chosen because the oversampled
result is inherently linear. The DAC is driven with apulse density modulatedsignal, created
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with the use of alow pass filter , step nonlinearity (the actual 1 bit DAC), and negative
feedbackloop, in a technique called delta sigma modulation . This results in an effective
high pass filter acting on the quantization (signal processing) noise, thus steering this noise
out of the low frequencies of interest into the high frequencies of little interest, which is
called noise shaping (very high frequencies because of the oversampling). The quantization
noise at these high frequencies are removed or greatly attenuated by use of an analog low
pass filter at the output (sometimes a simple RC low pass circuit is sufficient). Most veryhigh resolution DACs (greater than 16 bits) are of this type due to its high linearity and low
cost. Higher oversampling rates can either relax the specifications of the output low pass
filter and enable further suppression of quantization noise. Speeds of greater than 100
thousand samples per second (for example, 192kHz) and resolutions of 24 bits are attainable
with Delta Sigma DACs. A short comparison with pulse width modulation shows that a1 bit
DAC with a simple first order integrator would have to run at 3 THz (which is physically
unrealizable) to achieve 24 meaningful bits of resolution, requiring a higher order low pass
filter in the noise shaping loop. A single integrator is a low pass filter with afrequency
response inversely proportional to frequency and using one such integrator in the noise
shaping loop is a first order delta sigma modulator. Multiple higher order topologies (such asMASH) are used to achieve higher degrees of noise shaping with a stable topology.
the Binary Weighted DAC, which contains one resistor or current source for each bit of the
DAC connected to a summing point. These precise voltages or currents sum to the correct
output value. This is one of the fastest conversion methods but suffers from poor accuracy
because of the high precision required for each individual voltage or current. Such high
precision resistors and current sources are expensive, so this type of converter is usually
limited to 8 bit resolution or less.
the R 2R ladder DAC, which is a binary weighted DAC that uses a repeating cascaded
structure of resistor values R and 2R. This improves the precision due to the relative ease of
producing equal valued matched resistors (or current sources). However, wide convertersperform slowly due to increasingly large RC constants for each added R 2R link.
the Thermometer coded DAC, which contains an equal resistor or current source segment
for each possible value of DAC output. An 8 bit thermometer DAC would have 255
segments, and a 16 bit thermometer DAC would have 65,535 segments. This is perhaps the
fastest and highest precision DAC architecture but at the expense of high cost. Conversion
speeds of >1 billion samples per second have been reached with this type of DAC.
Hybrid DACs, which use a combination of the above techniques in a single converter. Most
DAC integrated circuits are of this type due to the difficulty of getting low cost, high speed
and high precision in one device.
the Segmented DAC, which combines the thermometer coded principle for the mostsignificant bits and the binary weighted principle for the least significant bits. In this
way, a compromise is obtained between precision (by the use of the thermometer
coded principle) and number of resistors or current sources (by the use of the binary
weighted principle). The full binary weighted design means 0% segmentation, the
full thermometer coded design means 100% segmentation.
DAC performance
DACs are at the beginning of the analog signal chain, which makes them very important to system
performance. The most important characteristics of these devices are: Resolution: This is the number of possible output levels the DAC is designed to reproduce.
This is usually stated as the number ofbits it uses, which is the base twologarithm of the
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number of levels. For instance a 1 bit DAC is designed to reproduce 2 (21) levels while an 8
bit DAC is designed for 256 (28) levels. Resolution is related to the Effective Number of
Bits (ENOB) which is a measurement of the actual resolution attained by the DAC.
Maximum sampling frequency: This is a measurement of the maximum speed at which the
DACs circuitry can operate and still produce the correct output. As stated in the Nyquist
Shannon sampling theorem, a signal must be sampled at over twice thefrequency of thedesired signal. For instance, to reproduce signals in all theaudible spectrum, which includes
frequencies of up to 20 kHz, it is necessary to use DACs that operate at over 40 kHz. The
CD standard samples audio at 44.1 kHz, thus DACs of this frequency are often used. A
common frequency in cheap computer sound cards is 48 kHz many work at only this
frequency, offering the use of other sample rates only through (often poor) internal
resampling.
monotonicity: This refers to the ability of DACs analog output to increase with an increase
in digital code or the converse. This characteristic is very important for DACs used as a low
frequency signal source or as a digitally programmable trim element.
THD+N: This is a measurement of the distortion and noise introduced to the signal by the
DAC. It is expressed as a percentage of the total power of unwantedharmonicdistortion and
noise that accompany the desired signal. This is a very important DAC characteristic for
dynamic and small signal DAC applications.
Dynamic range: This is a measurement of the difference between the largest and smallest
signals the DAC can reproduce expressed in decibels. This is usually related to DAC
resolution and noise floor.
Other measurements, such asPhase distortion and Sampling Period Instability, can also be very
important for some applications.
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