303-group Abdukarimov Jokhongir laboratory work №1 study of the spectra of periodic signals

303-group Abdukarimov Jokhongir

LABORATORY WORK №1

STUDY OF THE SPECTRA OF PERIODIC SIGNALS

Purpose of the work: To study the amplitude and phase spectra of periodic signals: rectangular pulses; alternating impulses; sawtooth pulses. Investigate the influence of signal parameters (amplitude, pulse repetition frequency, shift of the signal position relative to the origin in% of the period) on the amplitude and phase spectra of periodic signals.

Test questions

1. 
   Give a definition of the concept of a periodic signal?
   

A periodic signal is one that repeats the sequence of values exactly after a fixed length of time, known as the period. A periodic signal is one that repeats the sequence of values exactly after a fixed length of time, known as the period. In mathematical terms a signal x(t) is periodic if there is a number T such that for all t Equation 10.10 holds.

(10.10)x(t)=x(t+T)

The smallest positive number T that satisfies Equation 10.10 is the period and it defines the duration of one complete cycle. The fundamental frequency of a periodic signal is given by Equation 10.11.

(10.11)f=1T

It is important to distinguish between the real signal and the quantitative representation, which is necessarily an approximation. The amount of error in the approximation depends on the complexity of the signal, with simple waveforms, such as the sinusoid, having less error than complex waveforms.

A non-periodic or aperiodic signal is one for which no value of T satisfies Equation 10.11. In principle this includes all actual signals since they must start and stop at finite times. However, aperiodic signals can be presented quantitatively in terms of periodic signals.

Examples of periodic signals include the sinusoidal signals and periodically repeated non-sinusoidal signals, such as the rectangular pulse sequences used in radar.

Non-periodic signals include speech waveforms and random signals arising from unpredictable disturbances of all kinds. In some cases it is possible to write explicit mathematical expressions for non-periodic signals and in other cases it is not.

In addition to periodic and non-periodic signals are those signals that are the sum of two or more periodic signals having different periods. T will not be satisfied in Equation 10.10, but the signal does have many properties associated with periodic signals and cannot be represented by a finite number of periodic signals.

2. 
   Name several physical processes for which the periodic signal model is a fairly accurate description.
   

Modulation is also a technical term to express the multiplication of the original signal by another, usually periodic, signal.

In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the carrier signal, with a modulating signal that typically contains information to be transmitted. The term analog or digital modulation is used when the modulating signal is analog or digital, respectively. Most radio systems in the 20th century used so-called analog modulation techniques: frequency modulation (FM) or amplitude modulation (AM) for radio broadcast since the original signal was analog. Most, if not all, modern transmission systems use QAM (Quadrature Amplitude Modulation) which changes the amplitude and phase of the carrier signal. As the modulating signal is a sequence or stream of bit, i.e., a digital modulating signal, the term digital modulation is used. However, it must be pointed out that, usually, the sequence of bits must be converted to an analog signal prior to the modulation (multiplication) by the carrier signal.

3. 
   How does the concept of negative frequency arise?
   

Perhaps the most well-known application of negative frequency is the calculation:

{\displaystyle X(\omega )=\int _{a}^{b}x(t)\cdot e^{-i\omega t}dt,}which is a measure of the amount of frequency ω in the function x(t) over the interval (a, b). When evaluated as a continuous function of ω for the theoretical interval (−∞, ∞), it is known as the Fourier transform of x(t). A brief explanation is that the product of two complex sinusoids is also a complex sinusoid whose frequency is the sum of the original frequencies. So when ω is positive, {\displaystyle e^{-i\omega t}} causes all the frequencies of x(t) to be reduced by amount ω. Whatever part of x(t) that was at frequency ω is changed to frequency zero, which is just a constant whose amplitude level is a measure of the strength of the original ω content. And whatever part of x(t) that was at frequency zero is changed to a sinusoid at frequency −ω. Similarly, all other frequencies are changed to non-zero values. As the interval (a, b) increases, the contribution of the constant term grows in proportion. But the contributions of the sinusoidal terms only oscillate around zero. So X(ω) improves as a relative measure of the amount of frequency ω in the function x(t).

4. 
   What properties does the spectral density of a real signal have?
   

Power spectral density function (PSD) shows the strength of the variations(energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak. The unit of PSD is energy per frequency(width) and you can obtain energy within a specific frequency range by integrating PSD within that frequency range. Computation of PSD is done directly by the method called FFT or computing autocorrelation function and then transforming it.

5. 
   How is it customary to determine the duration of impulse signals?
   

Impulse signals are defined as δ(t) = 0 for t ≠ 0 and ∫−∞+∞δ(t)dt=1 for the continuous time case, and δ(n)={1n=00n≠0 for the discrete time case.

6. 
   What is the relationship between the pulse duration and the width of its spectrum?
   

There are actually different definitions of a pulse duration:

• 
  The most frequently used definition is based on the full width at half-maximum (FWHM) of the optical power versus time. This is not sensitive to some weak pedestals as often observed with optical pulses.
  
• 
  For calculations concerning soliton pulses, it is common to use a duration parameter τ which is approximately the FWHM duration divided by 1.76, because the temporal profile can then be described as a constant times sech²(t / τ).
  
• 
  For complicated pulse profiles, a definition based on the second moment of the temporal intensity profile is more appropriate.
  
• 
  Particularly in the context of laser-induced damage, one sometimes uses an effective pulse duration, which is defined as the pulse energy divided by the peak power
  

7. 
   How the derivation operations are displayed in the frequency domain
   

and signal integration?

In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.^[1] Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a sum or integral of sine waves of different frequencies, each of which represents a frequency component. The "spectrum" of frequency components is the frequency-domain representation of the signal. The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain.

Some specialized signal processing techniques use transforms that result in a joint time–frequency domain, with the instantaneous frequency being a key link between the time domain and the frequency domain.

8. 
   How are the spectral densities of a video pulse and a radio pulse related to each other?
   
9. 
   What is the meaning of the concept of complex frequency?
   

A complex number used to characterize exponential and damped sinusoidal motion in the same way that an ordinary frequency characterizes simple harmonic motion; designated by the constant s corresponding to a motion whose amplitude is given by Ae^st, where A is a constant and t is time.

10. 
    Fast Fourier transform.
    

A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies.^[1] This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors.^[2] As a result, it manages to reduce the complexity of computing the DFT from {\displaystyle O\left(N^{2}\right)}, which arises if one simply applies the definition of DFT, to {\displaystyle O(N\log N)}, where {\displaystyle N} is the data size. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.