Fourth chapter of DSP

Fourier series are used in periodic signals, while Fourier transforms are applied in periodic and aperiodic signals. It is also possible to cycle the signal with Fourier series, while the energy signal uses the Fourier transform (because the energy signal is non-periodic except for all 0). Fourier transform/progression is just an analytical tool that is used to analyze

Spectrum of. Spectrum generally refers to the frequency content of the signal, spectrum analysis is theoretical analysis, spectrum estimation is the actual measurement and re-judgment, so there will be a certain error.

The advantage of decomposing the original signal into a positive cosine or complex exponential function is that the two basic unit signals are unchanged by the LTI system frequency and can only change amplitude or phase.

The period of one domain corresponds to the discretization of another domain.

The periodic signal satisfies the conditions of the DI, the Fourier series can be used. The endrin condition is only sufficient condition that the signal can be expressed in Fourier, that is, there are some signals that do not meet the conditions of the DI, but can still be expressed by Fourier series.

The general actual signal satisfies the three conditions of endrin.

The real signal spectrum satisfies the conjugate symmetry.

There are three kinds of Fourier series in the real period signal, which is mentioned in this book, but it is not very good. These three types are: the first is referred to by the complex index, the second is the learning signal and the system first contact with the trigonometric function of the positive cosine of the form, plus a mean value component. This is not the same as the first thing to do with X (t) multiplied, the scale outside the integral is 2/t, not 1/t. The third expression is introduced by the first expression, the use of the real signal spectrum conjugate symmetry of the nature of the two positive and negative frequencies corresponding to the merger, of course, plus the same as the second DC component. Xn=0.5an-j*0.5bn.

Because the periodic signal belongs to the power signal, it is possible to study the distribution of power in the frequency domain, and there is a power spectral density (PSD) one said. The amplitude of the PSD of the continuous periodic signal is the square of each frequency coefficient, and the amplitude of all frequencies is added to the average power.

The Singh function Sinc=sinpix/pix. The last amplitude is gradually attenuated to 0.

When the continuous periodic function is a real-even function, the Fourier coefficients are real numbers. This is because according to the Fourier coefficient of the formula, in the integral inside every T-T, the imaginary part of the complex number is eliminated (the book is not written is a real function, but the whole will know that it should be a real function)

When a complex number is a real number, it can only have two really different phases 0 and pi. So the amplitude spectrum and the phase spectrum to be depicted can be integrated into a picture. The spacing of discrete spectral lines is only related to the period of the original signal, and when the period is infinite, the spacing between lines tends to be 0.

For the continuous-time non-periodic signal, the most studied is the non-periodic energy signal, the most typical is a signal (finite length). The solution is to extend the advanced period and then approximate the period to infinity.

The Fourier transform form of a continuous-time non-periodic signal is very simple, and there is no periodic signal that is multiplied by the coefficients.

The energy is limited by the absolute integral can be introduced, but not the other way. For example, the singer function, this is because the amplitude squared later rather smaller reasons.

The signal of limited energy is of course to study the distribution of energy in the frequency domain, that is, the energy spectrum. Obviously the energy spectrum of the real signal is symmetrical.

The more the time domain expands, the more concentrated the frequency domain.

The difference between the discrete time signal and the continuous time signal is that its frequency component is limited, the frequency domain is continuous is-pi to the PI, the frequency domain is discrete is the maximum one period n.

The Fourier series (DTFS) of discrete time-period signals is discussed first, because it is a periodic signal, so the frequency domain is also discrete and has n components. The time domain is discrete, which causes the frequency domain to be periodic. And the cycle is also n

The discrete-time signal is obviously also considered power spectrum (the power here is the same as the previous, average power). The amplitude is also the square of the coefficient modulus.

Fourier coefficients have a property of ck=ck-n=cn-k conjugate. So the n coefficients are all mastered by the first half of the information.

Discrete non-periodic spectrum is continuous, but only limited to 2pi intervals.

Verify the absolute integrals first when validating the Fourier transform

The Fourier transform can be seen as a Z-transform on a unit circle. But the premise is that the unit circle is within the convergent domain.

If a non-periodic sequence is neither absolute integrable nor square integrable, the Fourier transform does not exist. It is possible to introduce an impulse function in the frequency domain.

Bandwidth and carrier frequencies are narrower than narrow-band signals. Band-limited signal. Time signal. A signal cannot be both a band-limited signal and a time-limit signal.

Real signal spectrum conjugate symmetry, real-even signal spectrum is real, real-odd signal spectrum is pure imaginary

Re on Re,io to Io,ro to Io,io to RO

290 is the various properties of DTFT

Practice:

4.1 is obviously a continuous periodic signal whose spectrum is discrete and non-periodic. The CK is obtained from the basic formula, and the impulse function is used to represent the form of continuous spectrum. The power time domain is averaged, and the Fourier series Squared Plus is added in the frequency domain. The Sin/cos function is multiplied by the complex exponential function, the trigonometric functions are represented by the complex exponent, and then the integral is much easier.

4.2 Continuous non-periodic function, its analysis and synthesis formula are the simplest, a set of use can be, to obtain the complex number and to obtain amplitude and amplitude angle

4.3 is also non-cyclical continuous, but the calculation is more troublesome. The way to feel the answer is good, but not to the level of proficiency ...

4.4 The period of first this series is 6, not 9 .... In addition, the general formula of CK to write, into a relatively simple form, and then bring into the value of k, or to each k in the more troublesome form counted once, so that can not be zoned. At the end of the Parseval's theorem validation time, the time domain is divided by n, and the frequency domain coefficients are squared plus.

4.5ck=ck+n Bashi Direct conversion of the power spectrum, power is the sum of the squares of each Fourier series.

4.6 A, B or use ck=ck+n c to convert each item and multiply it to get the same result as before.

D, Spectrum is not conjugate symmetric, not real signal

E, the question is to write the general formula of CK first.

F, first to find out the general formula of CK, and then to the inside algebra. The form of E^JA1+E^JA2 can be merged.

G,h for non-cyclical continuous signals, it must be the research scope if the energy is limited. So this can only be judged as a periodic signal, it is considered n=1 to be finished. Another n=2.

4.7

A, the calculation of the obvious exceeded the need to find a way

b, a bunch of 0 good deals

C, this general trickery, two items of merger.

4.8

A, calculated by the equal ratio formula, divided q=1 and not equal to the discussion

b, double-loop variable, specify one, change another

C, it's easy to launch.

4.9 ABCD KO

E to meet the conditions of absolute integrable, Fourier transform can be carried out

F,g,h KO

4.10 C D The answer to a real part of the symbol, in fact, no need, the final merge as can be done

4.11 The coupling of the signal produces the real part of the Fourier transform, and the singular part of the signal produces the entire imaginary part of the Fourier transform. Y (n) based on linear properties and time-lapse properties

4.12 Straight set of formulas, and the integral of the segment came out.

4.13 actually gave the answer directly ... is also a typographical error, using the linear properties of the Fourier transform

4.14 A is to find that the sum of the series and b real-even signals will produce real spectrum, the angle can only be pi. Because the value of Oumiga is open interval, so marry less than 0, this point in the previous calculation and other than the time of the sequence is also very important

C 2pix (0) d-9 e Parseval's theorem

4.15 The derivative of the trigonometric functions at 0 points should be 0, because even a tangent is a straight line parallel to the x-axis.

Fourth chapter of DSP