Chapter 2 - Discrete-time signals and systems

Discrete-time signals, energy and power of discrete signals, signal symmetries, block diagrams, classification, convolution.

Dubulicious 2024-03-25
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Examples of Discrete time signals

Discrete time signal


The underline indicates `n=0`.

Discrete time systems

Average of three last values of `x(n)`:


A stable system decaying at a rate of 0.9:


A unstable system diverging at a rate of 1.1:


Discrete time signals

The Dirac delta function is used for impulses.

$$\delta(n)=\begin{cases}1&\text{if }n=0\\0&\text{otherwise}\end{cases}$$

The Step function (Heaviside/unit).

$$u(n)=\begin{cases}1&\text{if }n\geq0\\0&\text{otherwise}\end{cases}$$

These two functions can be combined to make other signals.


Exponential signals


Energy and Power of discrete signals

The energy of a signal is defined as:


The average power of a signal is defined as:


Even if a signal has infinite energy, the average power could still be finite.

We can also express the energy as:




Energy Signal and Power Signal

$$\text{energy signal}\iff 0 < E < \infty,P=0$$

$$\text{power signal}\iff 0 < P < \infty,E=\infty$$

Signal symmetries

A signal is symmetric or even if `x(n) = x(-n)`.

A signal is antisymmetric or odd if `x(n) = -x(-n)`.

Any signal `x(n)` can be written as a sum of an odd and even signal.

x(n) &= x_e(n) + x_o(n) \\
x_e(n) &= \frac{1}{2}(x(n)+x(-n)) \\
x_o(n) &= \frac{1}{2}(x(n) - x(-n))

Block diagrams

For block diagrams see lecture slides or chapter 2.2.2 in the book.


We call systems different things depending on their properties, a system is called:

  • static: if without memory
  • dynamic: if without memory,
  • time-invariant: where I/O characteristics don't change with time,
  • time-variant: where I/O characteristics change with time,
  • linear system: when it satisfies the superposition principle,
  • non-linear system: when it doesn't satisfy the superposition principle,
  • causal: where the output at any time only depends on the current and past inputs,
  • bounded-input bounded-output (BIBO) stable: where bounded inputs `|x(n)| \leq M_x` always produce bounded outputs `|y(n)| \leq M_y`.

LTI systems

Focus is on Linear Time-Invariant (LTI) systems in this course.

$$\delta(n) \rightarrow \boxed{\text{LTI system}} \rightarrow h(n)$$

The impulse response `h(n)` of an LTI system is the output signal when the input signal is a unit impulse `\delta(n)` at time `n=0`.

Discrete time convolution

An arbitrary signal can be written as the sum of delayed impulses:

$$x(n) = \sum_{k=-\infty}^\infty x(k)\delta(n-k)$$

Since LTI systems are linear and time-invariant the output `y(n)` will be the sum of impulse responses:

$$y(n) = x(n)*h(n) = \sum_{k=-\infty}^\infty x(k)h(n-k)$$

Convolution properties

Unit impulse is the identity element of convolution:

$$x(n)*\delta(n) = \sum_{k=-\infty}^\infty x(k)\delta(n-k) = x(n)$$

Convolution fulfills the:

  • commutative law: \(x*h = h*x\),
  • associative law: \((x*h_1)*h_2 = x * (h_1*h_2)\),
  • distributive law: \(x*(h_1+h_2) = x*h_1 + x*h_2\)

Convolution length

If we have two finite-length signals `x(n)` and `h(n)` with lengths `M` and `N` then the convolution \(y(n) = x(n) * h(n)\) will have length `M+N-1`


An LTI system is only causal if its impulse response fulfills: `h(n) = 0` for all `n < 0`.


Constant-coefficient difference equations

If we take the example of a recursive system:

$$y(n) = ay(n-1)+x(n)$$

We can rewrite it in terms of the memory from `n=0`:

$$y(n) = a^{n+1}y(-1) + \sum_{k=0}^n a^kx(n-k)$$

Constant-coefficient difference equations describe LTI systems.

Zero-state response

If the state of the system at `n=0` is zero (relaxed): e.g. `y(-1)=0` we get the zero-state response:

$$y_{zs}(n) = \sum_{k=0}^na^kx(n-k)$$

Zero-input response

If the input signal is zero, `x(n) = 0`, we get the zero-input response (natural response):

$$y_{zi}(n) = a^{n+1}y(-1)$$

The output signal is the sum: `y(n) = y_{zi}(n) + y_{zs}(n)`.