Digital Signal Processing

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Fundamental Concepts of Digital Signal Processing

Signal Classification

Continuous-time signal: Analog signal, continuous in the time domain.
Discrete-time signal: Continuous in amplitude but discrete in time.
Amplitude-discrete signal: Discrete in amplitude but continuous in time.
Digital signal: Discrete in both amplitude and time.

Differences

The distinction between a discrete-time signal and a digital signal lies solely in the quantization error present in digital signals.

Implementation Methods of Digital Signal Processing

The primary target of digital signal processing is digital signals, achieved through numerical operations.

Software Implementation

Programs are written based on principles and algorithms and executed on general-purpose computers.

Advantages: Flexibility
Disadvantages: Slow computation speed, difficult to achieve real-time processing.
Suitable for: Algorithm research and simulation.

Hardware Implementation

Hardware structures are designed according to specific requirements and algorithms, using basic components such as multipliers, adders, delay units, controllers, memory, and input/output interfaces.

Advantages: Fast computation speed, capable of real-time processing.
Disadvantages: Inflexibility

Hardware implementation refers to selecting an appropriate DSP chip, equipped with suitable software and language for the task, to achieve specific signal processing functions.

Dedicated Chips

Using Digital Signal Processing (DSP) chips is currently the fastest-growing and most widely applied method. DSP chips offer significant advantages over general-purpose microcontrollers:

Internal multipliers and accumulators tailored for digital signal processing.
Pipeline architecture, parallel structures, and multiple buses.
Specialized instructions optimized for digital signal processing, enabling high-speed computation.

For ultra-high-speed real-time systems where DSP chips are insufficient, Field-Programmable Gate Arrays (FPGAs) or custom ASICs (Application-Specific Integrated Circuits) should be employed.

Characteristics of Digital Signal Processing

Compared to analog signal processing, digital signal processing offers:

Flexibility
High precision and stability
Ease of large-scale integration
Capability to perform functions unattainable by analog systems, such as storage, complex transformations, and operations.

Signal Dimensionality

A signal is typically a function of one or more independent variables.

One-dimensional signal: Only one independent variable.
Multidimensional signal: Two or more independent variables.

Discrete-Time Signals and Systems

Discrete-Time Signals

In practice, signals are generally analog. Uniform sampling (equally spaced sampling) converts them into discrete-time signals.

For an analog signal $ x_a(t) $, discrete time points $ t_n $.
With uniform sampling interval $ T $, $ t_n = nT $:

$$ x(n) = x_a(t) \big|_{t=nT} = x_a(nT), \quad -\infty \lt n \lt \infty $$

Here, $ x(n) $ is called a discrete-time signal, where $ n $ is an integer, forming a sequence:

$$ x(n) = \{\cdots, x_a(-2T), x_a(-T), x_a(0), x_a(T), x_a(2T), \cdots \} $$

Discrete-time signals are also referred to as sequences.

Representation Methods for Sequences

Set Notation

A set of numbers is denoted by $ \{\cdot\} $. A discrete-time signal can be represented as an ordered set of numbers.
The underlined element in the set indicates the sample value at $ n=0 $.

Formula Representation

Example:

$$ x(n) = a^{|n|}, \quad 0 \lt a \lt 1, \quad -\infty \lt n \lt \infty $$

Graphical Representation

The horizontal axis represents $ n $, and the vertical axis represents the value of $ x $, with dots atop vertical lines.

Common Standard Sequences

Unit Impulse Sequence $ \delta(n) $

$$ \delta(n) = \begin{cases} 1 & n = 0 \\ 0 & n \neq 0 \end{cases} $$

Also called the unit sample sequence, distinct from the unit impulse signal $ \delta(t) $.

Unit Step Sequence $ u(n) $

$$ u(n) = \begin{cases} 1 & n \geq 0 \\ 0 & n \lt 0 \end{cases} $$

Relationships:

$$ \delta(n) = u(n) - u(n-1) $$

$$ u(n) = \sum_{k=0}^{\infty} \delta(n - k) $$

Rectangular Sequence $ R_N(n) $

$$ R_N(n) = \begin{cases} 1 & 0 \leq n \leq N-1 \\ 0 & \text{Otherwise} \end{cases} $$

$ N $ is the length of the rectangular sequence. It can be expressed using the unit step sequence:

$$ R_N(n) = u(n) - u(n - N) $$

Real Exponential Sequence

$$ x(n) = a^n u(n), \quad a \text{ is a real number} $$

Convergent sequence: $ |a| \lt 1 $
Divergent sequence: $ |a| \gt 1 $

Sinusoidal Sequence

$$ x(n) = \sin(\omega n) $$

Here, $ \omega $ is the digital frequency (units: radians, $ rad $), representing the rate of change (phase shift between adjacent samples).

Analog angular frequency $ \varOmega $
If the sinusoidal sequence is derived from sampling an analog signal $ x_a(t) = \sin(\varOmega t) $:

$$ x(n) = x_a(t) \big|_{t=nT} = \sin(\varOmega nT) = \sin(\omega n) $$

The relationship between digital and analog frequencies is:

$$ \omega = \varOmega T $$

Given the sampling frequency $ F_s = \frac{1}{T} $:

$$ \omega = \frac{\varOmega}{F_s} $$

Digital frequency is the normalized analog angular frequency relative to the sampling frequency.

Complex Exponential Sequence

$$ x(n) = e^{(\sigma + j \omega_0)n} = \cos(\omega_0 n) + j \sin(\omega_0 n) $$

Since $ n $ is an integer, both sinusoidal and complex exponential sequences are periodic with period $ 2\pi $.

Periodic Sequence

If for all $ n $, there exists a smallest positive integer $ N $ such that:

$$ x(n) = x(n + N), \quad -\infty \lt n \lt \infty $$

then the sequence $ x(n) $ is periodic with period $ N $.

Sequence Operations

Addition and Multiplication

Shifting, Flipping, and Scaling

Discrete-Time Systems

For a system with input $ x(n) $, output $ y(n) $, and operation $ T[\cdot] $:

$$ y(n) = T[x(n)] $$

Linear Systems

A system is linear if its input-output relationship satisfies the principle of superposition.

Additivity

$$ y_1(n) = T[x_1(n)], \quad y_2(n) = T[x_2(n)] $$

$$ T[x_1(n) + x_2(n)] = y_1(n) + y_2(n) $$

Homogeneity (Scaling)

$$ T[a \cdot x(n)] = a \cdot y(n) $$

Time-Invariant Systems

A system is time-invariant if its operation $ T[\cdot] $ does not change over time, i.e., the system’s response is independent of when the input is applied.

$$ y(n) = T[x(n)] $$

$$ y(n - n_0) = T[x(n - n_0)] $$

Characteristics of Linear Time-Invariant (LTI) Systems

Total response = Zero-input response + Zero-state response

Unit Impulse Response

With zero initial state (no zero-input response):

$$ h(n) = T[\delta(n)] $$

For any input $ x(n) $:

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

Thus, the output is:

$$ \begin{align} y(n) &= T[x(n)] \\ &= T\left[\sum_{m=-\infty}^{\infty} x(m) \delta(n - m)\right] \\ &= \sum_{m=-\infty}^{\infty} x(m) T[\delta(n - m)] \\ &= \sum_{m=-\infty}^{\infty} x(m) h(n - m) \\ &= x(n) * h(n) \end{align} $$

This is the convolution sum. For details, refer to Signals and Systems.

Causality of Systems

Definition: A system is causal if its output at time $ n $ depends only on the input at time $ n $ and prior inputs, not future inputs.

Necessary and Sufficient Condition:
The unit impulse response satisfies:

$$ h(n) = 0 \quad \text{for} \quad n \lt 0 $$

Stability of Systems

Definition: A system is stable if every bounded input produces a bounded output (BIBO stability).

Necessary and Sufficient Condition:
The unit impulse response is absolutely summable:

$$ \sum_{m=-\infty}^{\infty} |h(n)| \lt \infty $$