6. Discrete-Time Systems

Question 1

What is a discrete-time system?

Answer 1

A discrete-time system is a digital processing unit that takes a (number of) discrete-time signal(s) at the input and converts them into a (number of) discrete-time signal(s) at the output, with as a main goal to enhance or to change the characteristics of the signal(s).

(p. 157)

Question 2

What is a linear system? Give a practical example of a linear discrete-time system.

Answer 2

A system is said to be linear if a linear combination of signals at the input gives rise to the same linear combination of corresponding signals at the output.

If y1[k] is the response of a linear system to signal x1[k] at the input, and y2[k] is the response to x2[k], then the response to the combined input ax1[k] + bx2[k] will be

                          ax1[k] + bx2[k] −→ ay1[k] + by2[k]              −∞ < k < ∞

If this property holds for all signals x1[k] and x2[k], and for all linear combinations a, b ∈ R, the system is said to be linear.

Practical examples are a linear digital amplifier or standard digital filters.

(p. 158)

Question 3

Give a practical example of a nonlinear discrete-time system.

Answer 3

Nonlinear discrete-time signals are either created on purpose offering some type of desired system behavior, e.g. audio/image/video compression (mp3, JPG, MPEG), dynamic range compression, pitch shifting, . . . , or they are designed to mimic a nonlinear real-life (analog) system, e.g. a model for a tube amplifier, a distillation column …

(p. 158)

Question 4

What is a time-invariant system? Give a practical example of a time-invariant and of a time-varying discrete-time system.

Answer 4

A system is said to be time invariant if the response to a signal at the input does not depend on when the input is applied. In other words, a time-invariant system is a system whose
characteristics do not change over time.

If the response to a signal at the input does not depend on when the input is applied. Hence, if y[k] is the response of a time-invariant system to input x[k], then y[k + K] will be the response to x[k + K], and this for all values of K ∈ Z and for all signals x[k].

The linear discrete-time amplifier y[k] = a · x[k] always produces the same output, regardless of when the input is applied.

In some discrete-time applications, however, time-varying behavior is explicitly aimed at, e.g. to create certain audio effects such as a phaser, or whenever an accurate digital model is required of a time-varying real-life system, such as the human voice production system, e.g. in the case of a voice synthesizer. The adaptive filter that is discussed in chapter 11 is another example of a time-varying discrete-time system.

(p. 159)

Question 5

What is a difference equation?

Note that equation 6.7 can be found in the formulary.

Answer 5

If a linear ordinary differential equation of the form of equation 6.2 is evaluated at t = kTs and if all the time derivatives are replaced with finite (backward) differences an expression similar to

                  formula 6.6

results. An equation of this type is called a difference equation. This illustrates that whereas (linear) continuous-time systems are described by (linear) ordinary differential equations, their discrete-time counterparts are modelled by (linear) difference equations.

Typically, equation 6.6 is normalized by dividing the left-hand part and the right-hand part of the equation by ¯a0 != 0, which leads to

                  formula 6.7

Equation 6.7 describes a general linear time-invariant discrete-time SISO system of order7 max(M, N) with input x[k] and output y[k].

(p. 161)

Question 6

What is a causal system?

Answer 6

In fact, the output of most real-life systems only depends on current and on previous inputs and outputs. Such systems are called causal systems.

(p. 161)

Question 7

What is the difference between a differential equation and a difference equation when it comes to performance?

Answer 7

Contrary to equation 6.2, equation 6.7 can be readily and efficiently implemented on a microprocessor, merely requiring basic operations like multiplications, additions and memory accesses.

(p. 161)

Question 8

What is a recursive discrete-time system?

What is a non-recursive discrete-time system?

Answer 8

The system defined by equation 6.7 is said to be recursive because output y[k] not only depends on (current and past) inputs x[k − n], but also on (previous) versions y[k − m] of itself. If the system output y[k] only depends on (current and past) inputs and not on (past) outputs, equation 6.7 reduces to

          formula 6.8

which represents the subclass of (causal) non-recursive linear discrete-time systems.

(p. 162)

Question 9

Make sure you understand what the different symbols and blocks in figure 6.2 stand for and how a block diagram can be implemented in software.

You should be capable of retrieving the difference equation from the block diagram. For instance, if you are given figure 6.3, you may be asked to find equations 6.9 and 6.10.

Answer 9

The oval structure labeled with a + sign is a summator, the triangles perform a multiplication by a scalar, and the rectangles with a ∆-sign inside are delay elements.

(p. 162 - 164)

Question 10

What is the impulse response of a discrete-time system? Make sure you can calculate the impulse response from the block diagram or from the difference equation.

Answer 10

The zero-state (also called zero-initial condition or zero-initial state) response of a discretetime system to a discrete Dirac impulse δ[k] at the input is called the impulse response h[k] of the system.

The impulse response of a (causal) system can be obtained by initially setting all the delay element values in the block diagram to zero, then applying a discrete Dirac impulse to the input, i.e. setting x[0] = δ[0] = 1 and x[k] = δ[k] = 0, k > 0, and recording the output y[k] = h[k].

(p. 163)

Question 11

What is a finite impulse response (FIR) system?

What is an infinite impulse response (IIR) system?

Answer 11

Based on the length of the impulse response, discrete-time systems can be classified as:

1. finite impulse response (FIR) systems, which are discrete-time systems with an impulse response h[k] that has finite length, i.e. h[k] is non-zero only at a finite number of time instances, as in
   the case of the first example system
2. infinite impulse response (IIR) systems, which are discrete-time systems with an impulse response h[k] that has infinite length, as in the case of the second example system
   (p. 165)

Question 12

What is certain when it comes to non-recursive discrete-time systems?

Answer 12

A non-recursive discrete-time system of (finite) order N always has an impulse response of finite length. The impulse response parameters correspond to the coefficients of the difference equation.

(p. 165)

Question 13

What is mostly the case when it comes to recursive discrete-time systems?

Answer 13

Recursive discrete-time systems almost always have an impulse response that is infinitely long. Hence, (most) recursive discrete-time systems are IIR systems.

(p. 166)

Question 14

What is the relationship between the impulse response and the input/output of the system?

Answer 14

The zero-state response y[k] of a linear time-invariant discrete-time system to a signal x[k] at the input is equal to the convolution of the input with the impulse response h[k] of the system, i.e. y[k] = h[k] ⋆ x[k].

(p. 166 - 167)

Question 15

What is the frequency response of a linear discrete-time system? Make sure you can calculate the frequency response from the impulse response.

Answer 15

The frequency response of a linear discrete-time system is the discrete-time Fourier transform (DTFT) of the impulse response. Hence, it follows from equation 4.12 that

          formula 6.20

is the frequency response of the linear discrete-time system with impulse response h[k]. As most systems are causal, i.e. h[k] = 0 for k < 0 (see section 6.4.1), equation 6.20 reduces to

          formula 6.21

(p. 167)

Question 16

What is the magnitude and the phase response of a linear discrete-time system?

Answer 16

Note that, typically, the frequency response is a complex-valued function of φ. Complexvalued functions can be represented using Cartesian (rectangular) or polar coordinates. Polar coordinates express the complex-valued function H(φ) as a magnitude response |H(φ)| and a phase response ∠H(φ) :

      formula 6.22

Recall that the magnitude response of a real-life system is an even function of φ and that the phase response is an odd function of φ. As the frequency response is furthermore periodic in φ = f/fs, it suffices to plot the response for values of φ going from 0 to 0.5.

(p. 167)

Question 17

What happens when an infinitely long (co)sine wave is applied to the input of a linear time-invariant system?

Answer 17

If an infinitely long (co)sine wave of frequency φ0 is applied to the input of a linear time-invariant system, also the output will be an infinitely long (co)sine wave of the same frequency. The amplitude will be multiplied by |H(φ0)| and the phase will be shifted forward by ∠H(φ0).

(p. 168 - 169)

Question 18

What will the Discrete-time Fourier Transform of the output of a linear time-invariant discrete-time system be if the initial state is zero?

Answer 18

If the initial state is zero, the discrete-time Fourier transform of the output of a linear time-invariant discrete-time system is the product of the discrete-time Fourier transform of the input and the frequency response of the system.

(p. 169)

Question 19

What is the z-transform?

Answer 19

The z-transform can be considered the discrete equivalent of the Laplace transform. It is applicable to discrete-time signals and can be used to solve (sets of) difference equations and to compactly describe discrete-time systems in the z-domain.

(p. 170)

Question 20

Make sure you understand formulas 6.38 and 6.40. Equation 6.38 can be found in the formulary.

Answer 20

(p. 171)

Question 21

What is the link between the z-transform and the discrete Fourier transform?

Answer 21

The discrete-time Fourier transform of a discrete-time signal x[k] can be obtained by evaluating the z-transform of x[k] on the unit circle, i.e. by replacing z with e^j2πφ.

(p. 171 - 172

Question 22

What is the time-shifting property of the (bilateral) z-transform? Explain in a few words.

The formula can be found in the formulary.

Answer 22

If x[k] bZT ←→ X(z) is a bilateral z-transform pair, then also

              x[k − K] bZT ←→ z^−K · X(z)

is a bilateral z-transform pair, where K ∈ Z. Hence, delaying signal x[k] by K samples comes down to multiplying the z-transform of x[k] by z^−K. It will become clear that this property is of great importance for the study of discrete-time systems and digital filters.

(p. 172 - 173)

Question 23

What does the convolution theorem state? The formula can be found in the formulary.
What is the importance of the convolution theorem?

Answer 23

If x[k] bZT ←→ X(z) and y[k] bZT ←→ Y (z) are bilateral z-transform pairs, then also

                  x[k] ⋆ y[k] bZT ←→ X(z) · Y (z) 

is a bilateral z-transform pair, where the discrete linear convolution operator ⋆ is defined by equation 4.27. The convolution theorem also holds for unilateral z-transform pairs provided that the signals are causal. Recall that similar properties exist for the Laplace transform and for the four Fourier transform variants that were considered in chapters 2 and 4. It turns out that in some applications it is more convenient or more efficient to compute equation 6.51 as a multiplication in the transform domain rather than as a computationintensive linear convolution operation in the time domain.

(p. 173)

Question 24

What is the transfer function of a linear discrete-time system?

Recall that the transfer function is given by H(z) = Y (z)/X(z).

Answer 24

The z-transform of the impulse response h[k] of a linear discrete-time system

    formula 6.53

is called the system function or the transfer function of the system. By applying the convolution property (equation 6.51) to equation 6.19 it is easily shown that

   formula 6.54

form a z-transform pair. The transfer function H(z) can therefore be found as the ratio of the z-transform of the system output to the z-transform of the system input (assuming zero initial state):

   formula 6.55

In other words, H(z) quantifies how input X(z) is transformed when it passes through the system, hence the name transfer function.

(p. 174 - 175)

Question 25

Make sure you can derive the transfer function from the impulse response as in equations 6.56 and 6.57.

Answer 25

(p. 175)

Question 26

Make sure you can find the transfer function (and hence the frequency response) from the difference equation and vice versa. In this respect, keep in mind that the coefficients of the transfer function can be directly derived from the underlying difference equation. Equations 6.7, 6.60 and 6.62 can be found in the formulary.

Answer 26

(p. 175 - 178)

Question 27

What is the effect of a factor z^-L in the numerator/denominator?

Answer 27

Recall that a factor z^−L in the numerator of H(z) delays the input signal x[k] by L samples and that a factor z^−L in the denominator delays the output signal y[k] by L samples.

(p. 178)

Question 28

What is a pole? What is a zero?

Answer 28

The poles of a linear discrete-time system are the roots of the transfer function denominator polynomial. In other words, they are the values of z that make the denominator of the transfer function H(z) zero, i.e. that let H(z) go to infinity.

The zeros of a linear discrete-time system are the roots of the numerator of H(z), i.e. those values of z that make the numerator of the transfer function zero, in other words, that let H(z) go to zero.

(p. 179)

Question 29

Make sure you can calculate the poles and the zeros of a transfer function.

Answer 29

To this end, write H(z) as a function of z, as in equation 6.62, and then calculate the roots of the denominator and the numerator polynomial. Keep in mind that you may need to solve a second-order polynomial equation of the type az2 +bz +c = 0, the solution of which is given by z = (−b±√(b2−4ac))/2a.

Question 30

What is the order of a linear discrete-time system?

Answer 30

Hence, the number of poles and zeros is equal to the order of the denominator and numerator polynomial, respectively. The maximum of those two is the order of the system.

(p. 179)

Question 31

What form do the poles and zeros of a real-life system have?

Answer 31

The poles and the zeros of a real-life system are real valued or appear in complex-conjugate pairs.

(p. 179)

Question 32

What is the zero-pole-gain representation of a linear discrete-time system? Equation 6.69 can be found in the formulary.

Answer 32

A zero-pole-gain representation offers a full description of the zero-state input-output behavior of the system.

H(z) is written as a product and ratio of an overall gain b0 and of first-order transfer functions containing the poles and the zeros. From this description it is clear that the poles pm make the denominator polynomial zero, and that the zeros bring the numerator to zero.

(p. 180)

Question 33

Make sure you can represent the poles and the zeros in a pole-zero diagram.

Answer 33

Poles and zeros are typically plotted in a so-called pole-zero diagram, showing the position of the poles and the zeros in the complex plane. Poles are commonly represented by ’×’, and zeros by ’◦’.

(p. 180)

Question 34

Under which condition is a causal discrete-time system stable?

Answer 34

A causal linear discrete-time system is stable if all the poles lie inside the unit circle.

(p. 182)

Question 35

non-recursive linear discrete-time systems vs. stability

Answer 35

Recall that non-recursive discrete-time systems are always stable.

(p. 182)

Question 36

What do the location of the poles/zeros influence?

Answer 36

Keep in mind that the location of the poles determines the dynamic properties and the stability of the system. The location of the zeros does not influence stability, but has an effect on the dynamic and spectral properties of the system.

(p. 183)