Constant, Sinusoid, Square, Triangle, and sawtooth waveforms, in depth and summarized.
Introduction
Once one has obtained a solid understanding of the fundamentals of
Fourier series
analysis and the
General Derivation of the Fourier Coefficients , it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.
Deriving the fourier coefficients
Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as
Real even signals
Given that the square wave is a real and even signal,
EVEN
*
REAL
therefore,
EVEN
* REAL
Consider this mathematical question intuitively: Can a
discontinuous function, like the square wave, be expressed as asum, even an infinite one, of continuous signals? One should at
least be suspicious, and in fact, it can't be thusexpressed.
The extraneous peaks in the square wave's Fourier series
never disappear; they are termed
Gibb's phenomenon after the American physicist
Josiah Willard Gibbs. They occur whenever the signal isdiscontinuous, and will always be present whenever the signal
has jumps.
Deriving the fourier coefficients for other signals
The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.
Constant waveform
This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.
Sinusoid waveform
With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.
Triangle waveform
This is a more complex form of signal approximation to the square wave. Because of the
Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that
ODD
*
REAL
therefore,
* IMAGINARY
Sawtooth waveform
Because of the
Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life