"Signal and System Study Notes" - Time Domain and Frequency Domain Characteristics of Signals and Systems (2)

Note: This blog is based on the second edition of Oppenheim's "Signals and Systems", mainly for the review and deepening of their own learning.

First, first-order and second-order continuous-time systems

1), first-order continuous-time system

1. For a first-order system, its differential equation is often expressed in the following form:

where t is a coefficient. The frequency response of the corresponding first-order system is

Its unit impulse response is

The step response of the system is

These are received in Figure 6.19) a) and Figure 6.19 (b) respectively. The parameter t is called the time constant of the system, which controls how fast the first-order system responds.

2. Figure 6.20 shows the Bode plot of the frequency response of equation (6.22). This graph demonstrates one of the advantages of using logarithmic frequency coordinates, which is that it is possible to obtain a very good approximation of a Bode diagram for a Deau-Austrian first-order system without much difficulty.

For this reason, the logarithmic characteristic of the frequency response, can be obtained by Eq. ( 6.22 )

It can be seen from this formula that for wt<<1, the logarithm is approximately zero; and for wt>>1, the logarithmic modulus is approximately a linear function of . That is to say

and

In other words, the asymptotes of the logarithmic model characteristic of a first-order system are straight lines at the first and high frequencies. The low-frequency asymptote [given by (6.26)] is a 0dB line; and the high-frequency asymptote [given by (6.27)] corresponds to 20dB every decade on |H(jw)| Attenuation, something this becomes a "20dB without a decade" asymptote.

Note that the two asymptotes represented by equations (6.26) and (6.27) are equal at this point, ie, w=1/t. From the graph, this means that the two asymptotes should intersect at w=1/t, thus providing a straight line approximation to the logarithmic model characteristic graph. That is, for w≤1/t, and for w≥1/t, it is given by equation (6.27).

Since the slope of the approximate characteristic changes at w=1/t, this point is often referred to as the corner frequency. At the same time, it can be seen from equation (6.25) that at w=1/t, the two terms in the logarithm are equal, so the actual value at this point is

For this reason, the point of w≥1/t is sometimes referred to as the 3dB point.

3. A useful straight line approximation can also be obtained as

It can be noted that this approximate property as a function of

The range is linearly decreasing (from 0 to -π/2), that is, there is a range of a decade above and below the corner frequency. Meanwhile, when w ≥ 1/t, the exact approximation is -π/2. Furthermore, at the corner frequency w = 1/t, the approximate value of the value is consistent with the true value, and its value is

4. The inverse relationship between time and frequency can be seen again from the first-order system above. When t decreases, the time response of the system is accelerated, that is, h(t) becomes more compressed toward the origin, and the rise time of the step response decreases; at the same time, the turning rate increases, that is , the frequency range becomes wider. wide, H(jw) becomes wider.

2), second-order continuous-time system

1. The general form of a linear constant coefficient differential equation for a second-order system is

The frequency response of its second-order system is

After factoring the denominator of H(jw), we get

If , by partial fraction expansion, we get

From equation (6.35), the unit impulse response of the system is

if this has

The unit impulse response at this time is

The parameter is called the damping coefficient and is called the undamped natural frequency. First of all, it can be seen from equation (6.35) that when 0 < < 1, both c1 and c2 are complex numbers, so the unit impulse response of equation (6.37) is written as

So for 0 < < 1, the unit impulse response of the second-order system is a cheap oscillation. The system is then said to be underdamped. If > 1, then both c1 and c2 are real and negative, and the unit impulse response is the difference between the two decaying exponents, and the system is called overdamped. When =, c1 = c2, then the system is called street-damped.

4. For ≠1, the step response of the second-order system can be calculated by equation (6.37), and its expression is

For =1, using equation (6.39), we can get

5. From formula (6.33), we can get

From this expression, we can find that the two linear asymptotes of the first frequency are

Therefore, the low-frequency asymptote of the logarithmic model characteristic is the 0dB line, while the high-frequency asymptote has a slope of -40dB every decade; that is, when w does not increase by a factor of 10, |H( jw)| will drop by 40dB. Also, the two asymptotes intersect at w=wn. Therefore, for w≤wn, the approximation given by equation (6.44) can be used to obtain a linear asymptote approximation for the logarithmic model characteristic. For this reason, wn is called the corner frequency of the second-order system.

In addition, the exact expression of a straight line approximation that can also be obtained can be obtained from equation (6.33)

The approximation of the pair is

3) Bode plot of rational frequency response

1. For a frequency response with the following form

and

Bode plot of , can be obtained quickly, because

and

At the same time, for the system whose system function is constant gain

Because, if K>0 ; if K<0 , then

Because a rational frequency summer camp can be factored into a product of a constant gain of 1 and the first-order and second-order terms, its Bode diagram can be obtained by adding the Bode diagrams of each time in the product.

Second, first-order and second-order discrete-time systems

1. Consider a first-order causal linear time-invariant system described by the following difference equation

where |a|<1. The frequency response of the system is

Its unit pulse is

Meanwhile, the step response of the system is

The modulus |a| of the parameter a is very similar to the effect of the time constant t in the continuous-time first-order system, that is, |a| determines the response rate of the first-order system. It is worth noting that, different from the first-order continuous-time system, the first-order system obtained by Eq. (6.51) can exhibit the characteristics of oscillation. This occurs when a<0, in which case the step response exhibits both excess and oscillatory characteristics.

2. The mode and phase of the frequency response of the first-order system described by Eq. (6.51) are respectively

and

When a>0, the system exhibits the characteristics of high frequency attenuation, that is, the value of |H(ejw)| when w is close to -+π is smaller than that when w is close to 0; 2. The frequency components are larger, and the low frequency components are attenuated. At the same time, it is also noted that for smaller values of |a|, the maximum value of |H(ejw)| 1/(1+a) and the minimum value of 1/(1-a) are gradually close in value, so | The change of H(ejw)| is relatively flat. On the other hand, when |a| is close to 1, the two values are quite different, and |H(ejw)| exhibits a steeper peak, which provides a good choice in a wider frequency band filtering and amplification.

2), second-order discrete-time system

1. Consider a second-order causal linear invariant system whose difference equation is

where 0<r<1, the frequency response of the system is

The denominator of the above formula can be factored to get

1) When it is not equal to 0 or π, the two factors are different, and the partial fraction expansion can be used to obtain

The unit impulse response of the system is then

2) When equal to 0 or π, the sum of the denominator of formula (6.58) and the two factors are the same. When = 0,

and

When = π,

and

It can be seen that the decay rate of h[n] is controlled by r, that is, the closer r is to 1, the slower h[n] decays. Similarly, the value determines the oscillation frequency. The different r and value effects can also be seen from the step response of Eq. (6.57).