DSP Chapter II

The discrete-time signal is not defined at all in the location of the sampling point interval and cannot be simply considered as a value of 0.

On the time series axis, move left called ahead, like right move called delay.

Down-sampling operation: fewer points are mined. For example, Y (n) =x (2n) is equivalent to only an even sample of X (n) is taken out.

The input of the system is called excitation, and the output of the system is called response.

The system has no excitation until the n0 time, and the system output is always 0, which is the initial relaxation condition. By default all systems in ancient times, t equals negative infinity, are initially slack.

The system will generate a signal.

Prove that a system does not have a certain nature, find a counter-example can. But to prove a certain nature, all signals must be established.

Static systems are memory-only and are related to the current input. Dynamic systems have memory, finite memory is a finite number of previous values coherent, infinite memory is associated with an infinite number of previous values.

Obviously there is no memory element in the static system.

When validating the LTI system, Y (N-K) replaces only n in the X (n) n-k substitution of N. Y (n,k) is the transformation of the system itself to n in x (n)

A loose, linear system. 0 inputs can produce 0 outputs. If an initial relaxation system does not meet the nonlinearity, it is a nonlinear system

To be able to pass a linear test system is called a linear system, the linear system to meet the initial relaxation, if dissatisfied with the initial relaxation, such as the n=0 when the input is 0, but the output is not 0, it is not the initial relaxation.

Offline systems tend to mean non-timely.

The series system is multiplied and the parallel system is added. Both systems in series are LTI and can be swapped in order

An input signal is represented as a time shift and a linear combination of basic element signals, typically expressed by a set of time-shifted impulse functions.

Because convolution generates input at various times, and the causal system is only relevant to current and past inputs, the future input values are eliminated from the system's point of view, i.e. h (n) =0 n<0

A sequence of n<0 with a value of 0 is called a causal sequence.

Linear time-invariant systems are divided into two categories: FIR (finite duration impulse response), Infinite duration impulse response (IIR)

Due to the form of convolution when H (k) in the infinite interval length is not 0 o'clock, the required input storage units are numerous and cannot be physically implemented. If it is done, it can only be described by the difference equation (with initial seed and recursive relationship can record the entire range)

In a recursive system, the value of the output y (n) of a system at n time depends on Y (n-1), Y (n-2) .... Then the system is called a recursive system.

The initial condition is the seed.

The sign of a recursive system is a feedback path.

The system described by the linear constant difference equation is a subclass of the recursive system and the non-recursive system.

A 0 state response is a response with an initial state of 0 (seed 0), and for an initial non-slack system, a response is generated even if no excitation is entered. A response that is not entered is called a 0 input response or a natural response. 0 The state response is called a forced response.

The system is linear to meet three conditions: 1, total response equals zero state response and 0 input response and 2, 0 input response satisfies superposition principle 3, 0 State response satisfies superposition principle

The systems described by linear constant difference equations are both linear and time invariant. The linearity is proven in three conditions, since the coefficients are constant, so it is time invariant.

0 The sum of the input response and the 0 state response is just a form of expression, not a specific solution.

The steady state response and the transient state response are not 0 when n approaches infinity, and the transition state response is 0 when n is approaching infinity.

Homogeneous uses initial conditions to determine constants.

FIR and IIR are different types of LTI systems, and recursive and non-recursive systems are the implementation methods of the system.

Correlation refers to the amount of information that a signal contains in another signal.

Correlation is a class of mathematical operations in which the object is to two sequences and then produces a new sequence. The two sequences are the same and are self-related. The difference is cross-correlation.

2.61 One is the given real sequence, the other is the energy signal, so there is a cumulative process.

The FIR system can be implemented recursively or non-recursive, and IIR is only implemented by recursive method.

After-school questions:

2.1 The difference between flipping after translation and flipping before panning

2.2 Makes Y (n) equal to f (x (n)) and the N value into X (n) for different y (n). The index of the odd part and the other. The answer e should be wrong, n=2 value is 0,n=3.

2.3 The n is divided into three intervals by 0, and the obtained values are exactly the same as each other.

2.4 The first card has, the re-certification of uniqueness. There is a simple, unique x (n) =y1 (n) +z1 (n), X (-N) =y1 (n)-z1 (-N). The solution is Y1 (n), Z1 (N). With understanding Y2 (n), Z2 (n). Discover the same, that is the only.

2.5 can be according to the answer so evidence, can also be specific expression of the second and odd part of the re-certification.

2.6 A certificate when the same, Y (n-k) very good proof, all the n is replaced by N-k can be. T{x (N-K)} only operates on N, not n-k overall. So here one is (n-k) ^2 The other is (N^2-K)

b y (n) =t (x (n)) n takes c from the outside to the inside, obviously time-invariant d.

2.7 Static systems are concerned only with current inputs (also known as memory-free systems), which are called dynamic systems associated with past or future values; linear is easy, time-invariant, and causality is only related to current or past inputs; the output is stable and the outputs are stable to prove the stability of the system.

The X (-n+2) of the D term should be a time-varying signal. H should be time-variant and L should be time-variant. n i am confused, do not belong to the discrete system, how to judge???

2.8LTI systems can be used for convolution characterization, convolution can be exchanged in order, so the cascaded LTI system may be swapped for positions.

I find the inverse example, the first rise and then drop the model, the first Infinity amplification, and then infinitely reduced, and finally only enlarge the constant number of times. Two nonlinear can produce a linear, such as Cos and Arccos. how is f found?

2.9 A, constant C, the stable system does not give infinite energy, so the input energy is limited, the output energy is also limited. b not the whole.

2.10 Easy

The 2.11 is output by linear input and is not changed when pushed. The first idea is to superimpose the two input signals into the time shift of the third signal to see if the output of the new signal satisfies the time-lapse relationship with the output of the third signal. But found no.

Here exactly the first two of the sum is the impulse function, if it is LTI to get convolution, take X3 to verify.

2.12 Easy

2.13 convolution to the inside of the set to get

2.14a didn't ask for it.

b convolution expression plus cause and effect is done.

2.15 Basic Geometric Series formula

2.16B Basic convolution operation A In doubt

2.17/18/19/21/22easy

2.20 Interesting

Relationship between the 2.23 step response and the impact response

The 2.24 coefficients have the very number must be time-varying, the output is C unbounded

2.25 See the answer to think out, 2,3 asked actually have clues back to see

2.26 input grinding into 0, using Lamda to calculate, back to the coefficient or to the original equation of x into 0

The 2.27 input is U (n) and does not guarantee initial slack.

2.28 the steady state response is the same as N to infinity, whereas the transition state to 0 is the corresponding

2.29 Easy

2.30 Note that the excitation and homogeneous solution form the same situation, at this time the excitation as a heavy root.

2.31 for H (N), the special solution is 0, the input impact function, 0 state corresponding (initial relaxation)

2.32easy

2.33 because the impact is the initial relaxation, so the first impulse corresponding, and then according to the relationship between the step response and impulse response to seek s (n)

2.34 take 8 points and you can count on it.

The 2.35LTI system cascades and concatenates, and the left arrow appears to be a symbol.

2.36easy

2.37 and convolution form control can be

2.38 sets of formulas

2.39/40/41/42easy

2.43 the sampling, screening and convolution properties of impulse functions are very important. how to prove that the system is slack?

2.44a,b is a reversible system

The 2.45 0 status response is equivalent to the initial condition of 0, which means that the output before the input starts is 0.

2.46 draw the second diagram of the implementation of the first kind of drawing out first.

2.47/48 Easy

2.49 How to find the inverse system of LTI

2.50confused is different from the examples in the book.

The calculation method of 2.51h (n) and the consistency of the book

2.52 for non-recursive sequences, the impulse response input impulse function can be obtained.

2.53confused is different from the examples in the book.

2.54 convolution and correlation sequence basic operation, the difference is the convolution must first reverse, the correlation sequence direct translation is good.

2.55 input signal is 0,m convolution signal is 0,n, output length is m+n-1

2.56

2.57 y (N) When you want to add the U (n), often homogeneous solution without U (n), Special solution plus U (n). Total plus U (n)

2.58 to H (n) also add U (n) plus (as with the method of example)

2.59 first, then a variable substitution

2.60 The first Ko, the second one.

2.61easy

The 2.62 autocorrelation function is even function, with a maximum value of 0 o'clock in the displacement.

2.63 The unit is divided by the self-correlation function of the displacement to 0

2.64 The maximum value of the autocorrelation function at a relative shift of 0 o'clock is used

DSP Chapter II