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Therefore, to avoid any aliasing or distortion of the discrete signal frequency content and to be able to recover or reconstruct the frequency content of the original analog signal, we must have f s 2f max size 12{f rSub { size 8{s} }>= 2f rSub { size 8{"max"} } } {} . This is known as the Nyquist rate. The sampling frequency should be at least twice the highest frequency in the analog signal. Normally, before any digital manipulation, a front-end anti-aliasing lowpass analog filter is used to limit the highest frequency of the analog signal.

Let us further examine the aliasing problem by considering an undersampled sinusoid as depicted in [link] . In this figure, a 1 kHz sinusoid is sampled at f s = 0 . 8 size 12{f rSub { size 8{s} } =0 "." 8} {} kHz, which is less than the Nyquist rate of 2 kHz. The dashed-line signal is a 200 Hz sinusoid passing through the same sample points. Thus, at the sampling frequency of 0.8 kHz, the output of an A/D converter is the same if one uses the 1 kHz or 200 Hz sinusoid as the input signal. On the other hand, oversampling a signal provides a richer description than that of the signal sampled at the Nyquist rate.

Ambiguity Caused by Aliasing


An A/D converter has a finite number of bits (or resolution). As a result, continuous amplitude values get represented or approximated by discrete amplitude levels. The process of converting continuous into discrete amplitude levels is called quantization. This approximation leads to errors called quantization noise. The input/output characteristic of a 3-bit A/D converter is shown in [link] to illustrate how analog voltage values are approximated by discrete voltage levels.

Characteristic of a 3-Bit A/D Converter: (a) Input/Output Transfer Function, (b) Additive Quantization Noise

Quantization interval depends on the number of quantization or resolution levels, as illustrated in [link] . Clearly the amount of quantization noise generated by an A/D converter depends on the size of the quantization interval. More quantization bits translate into a narrower quantization interval and, hence, into a lower amount of quantization noise.

Quantization Levels

In [link] , the spacing Δ between two consecutive quantization levels corresponds to one least significant bit (LSB). Usually, it is assumed that quantization noise is signal-independent and is uniformly distributed over –0.5 LSB and 0.5 LSB. [link] also shows the quantization noise of an analog signal quantized by a 3-bit A/D converter and the corresponding bit stream.

Quantization of an Analog Signal by a 3-Bit A/D Converter: (a) Output Signal and Quantization Error, (b) Histogram of Quantization Error, (c) Bit Stream

A/d and d/a conversions

Because it is not possible to have an actual analog signal within a computer programming environment, an analog sinusoidal signal is often simulated by sampling it at a very high sampling frequency. Consider the following analog sine wave:

x ( t ) = cos ( 1000 t ) size 12{x \( t \) ="cos" \( 2π"1000"t \) } {}

Sample this sine wave at 40 kHz to generate 0.125 seconds of x ( t ) size 12{x \( t \) } {} . Note that the sampling interval,seconds, is very short, so x ( t ) size 12{x \( t \) } {} appears as an analog signal.

Sampling involves taking samples from an analog signal everyseconds. The above example generates a discrete signal x [ n ] size 12{x \[ n \] } {} by taking one sample from the analog signal everyseconds. To get a digital signal, apply quantization to the discrete signal.

According to the Nyquist theorem, an analog signal z can be reconstructed from its samples by using the following equation:

z ( t ) = k = z [ kT s ] sin c ( t kT s T s ) size 12{z \( t \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {z \[ ital "kT" rSub { size 8{s} } } \] "sin"c \( { {t - ital "kT" rSub { size 8{s} } } over {T rSub { size 8{s} } } } \) } {}

This reconstruction is based on the summations of shifted sinc (sinx/x) functions. [link] illustrates the reconstruction of a sine wave from its samples achieved in this manner.

Reconstruction of an Analog Sine Wave Based on its Samples, f size 12{f} {} = 125 Hz and f s size 12{f rSub { size 8{s} } } {} = 1 kHz

It is difficult to generate sinc functions by electronic circuitry. That is why, in practice, one uses an approximation of a sinc function. [link] shows an approximation of a sinc function by a pulse, which is easy to realize in electronic circuitry. In fact, the well-known sample and hold circuit performs this approximation.

Approximation of a Sinc Function by a Pulse

Dtft and dft

Fourier transformation pairs for analog and discrete signals are expressed in [link] . Note that the discrete-time Fourier transform (DTFT) for discrete-time signals is the counterpart to the continuous-time Fourier transform (CTFT) for continuous-time signals. Also, the discrete Fourier transform (DFT) is the counterpart to the Fourier series (FS) for continuous-time signals as shown in [link] . [link] shows a list of these transformations and their behavior in the time and frequency domains.

Fourier series pairs for analog and digital signals
Fourier series for periodic analog signals { X k = 1 T T / 2 T / 2 x ( t ) e 0 kt dt x ( t ) = k = X k e 0 kt size 12{ left lbrace matrix { X rSub { size 8{k} } = { {1} over {T} } Int rSub { - T/2} rSup {T/2} {x \( t \) e rSup { size 8{ - jω rSub { size 6{0} } ital "kt"} } ital "dt"} {} ##size 12{x \( t \) = Sum cSub {k= - infinity } cSup { infinity } {X rSub {k} size 12{e rSup {jω rSub { size 6{0} } ital "kt"} }} } } right none } {} , where T denotes period andω 0 fundamental frequency
Discrete Fourier transform (DFT) for periodic discrete signals { X [ k ] = n = 0 N 1 x [ n ] e j N nk , k = 0,1, . . . , N 1 x [ n ] = 1 N k = 0 N 1 X [ k ] e j N nk , n = 0,1, . . . , N 1 size 12{ left lbrace matrix { X \[ k \]= Sum cSub { size 8{n=0} } cSup { size 8{N - 1} } {x \[ n \] e rSup { size 8{ - j { {2π} over {N} } ital "nk"} } } matrix {, {} # k=0,1, "." "." "." ,N - 1{} } {} ##x \[ n \] = { {1} over {N} } Sum cSub { size 8{k=0} } cSup { size 8{N - 1} } {X \[ k \]e rSup { size 8{j { {2π} over {N} } ital "nk"} } matrix { , {} # n=0,1, "." "." "." ,N - 1{}} } {} } right none } {}
Different Transformations for Continuous and Discrete Signals
Time domain Spectrum characteristics Transformation type
Continuous (periodic) Discrete FS
Continuous (aperiodic) Continuous CTFT
Discrete (periodic) Discrete (periodic) DFT
Discrete (aperiodic) Continuous (periodic) DTFT

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, An interactive approach to signals and systems laboratory. OpenStax CNX. Sep 06, 2012 Download for free at http://cnx.org/content/col10667/1.14
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