Introduction to Digital Signal Processing Flashcards

These flashcards cover the core vocabulary and foundational concepts of Digital Signal Processing (DSP), including signal types, LTI system properties, transforms (DTFT, DFT, z-transform), sampling theory, and filter design methodologies.

Continuous-time signals

Signals that are often referred to as analog signals, where the independent variable (typically time) is continuous.

Digital signals

Signals for which both the independent variable (time) and the amplitude are discrete.

Discrete-time systems

Systems where both the input and the output are discrete-time signals.

Sampling frequency ($f_s$)

The reciprocal of the sampling period, defined mathematically as $f_s = \frac{1}{T}$, expressed in units such as $Hz$ or radians per second as \text{\omega}_s = \frac{2\text{\pi}}{T}.

Unit sample sequence (\text{\delta}[n])

Also referred to as the discrete-time impulse or simply impulse, defined as \text{\delta}[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}.

Unit step sequence ($u[n]$)

A basic sequence defined as $u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}$.

Memoryless systems

Systems where the output $y[n]$ at any arbitrary time index $n$ depends only on the input $x[n]$ at that same value of $n$.

Linear systems

Systems defined by the principle of superposition, requiring the meet of both the additivity property and the scaling (homogeneity) property.

Time-invariant systems

Systems where a time shift or delay of the input sequence causes an identical corresponding shift in the output sequence.

Causal system

A system where the output at time index $n$ depends only on the values of the input at index $n$ and earlier time instants.

Stable system

A system that produces a bounded output sequence for every bounded input sequence ($BIBO$ stability).

Impulse response ($h[n]$)

The response of a linear system to a unit sample sequence input; it completely characterizes a Linear Time-invariant ($LTI$) system.

Convolution sum

The mathematical expression for the output of an $LTI$ system, denoted as $y[n] = x[n] * h[n] = \sum_{k = -\infty}^{+\infty} x[k]h[n - k]$.

$FIR$ systems

Finite-duration impulse response systems, characterized by an impulse response with a finite number of nonzero samples; these systems are always stable if the impulse response values are finite.

$IIR$ systems

Infinite-duration impulse response systems, where the response to an impulse lasts for an infinite duration of time.

Eigenfunction

A sequence $x[n]$ for which the system output is a scaled version of the same sequence, such as the complex exponential e^{j\text{\Omega}n} for $LTI$ systems.

Frequency response (H(e^{j\text{\Omega}}))

The eigenvalue of an $LTI$ system corresponding to the complex exponential input, defined as H(e^{j\text{\Omega}}) = \sum_{n = -\infty}^{+\infty} h[n]e^{-j\text{\Omega}n}.

Nyquist Sampling Theorem

A theorem stating that a bandlimited signal with \text{X}_c(j\text{\omega}) = 0 for |\text{\omega}| \text{\ge} \text{\omega}_N can be uniquely determined by its samples if the sampling frequency \text{\omega}_s \text{>} 2\text{\omega}_N.

Aliasing distortion

A type of distortion that occurs when the sampling frequency is not high enough (\text{\omega}_s \text{\le} 2\text{\omega}_N), causing overlapping copies of the Fourier transform.

Discrete Fourier Transform ($DFT$)

A Fourier representation for finite-duration sequences of length $N$ that corresponds to equally spaced samples of the $DTFT$, defined as \text{X}[k] = \sum_{n = 0}^{N - 1} x[n]e^{-j\frac{2\text{\pi}}{N}kn}.

Fast Fourier Transform ($FFT$)

A collection of efficient algorithms used to compute the Discrete Fourier Transform by decomposing the computation into smaller segments.

z-Transform

The discrete-time counterpart of the Laplace transform, defined for a sequence $x[n]$ as the power series $\text{X}(z) = \sum_{n = -\infty}^{+\infty} x[n]z^{-n}$.

Region of Convergence ($ROC$)

The set of values of the complex variable $z$ for which the z-transform sum converges.

Poles

The values of $z$ for which the z-transform $\text{X}(z)$ is infinite; for rational functions, these are the roots of the denominator polynomial.

Zeros

The values of $z$ for which the z-transform $\text{X}(z) = 0$; for rational functions, these are the roots of the numerator polynomial.

Direct Form II

A computational structure for implementing an $LTI$ system that rearranges the block diagram to share delay elements, minimizing the memory required.

Gibbs phenomenon

The oscillatory behavior that occurs near a discontinuity (like a brick-wall cutoff) in the frequency response when an ideal impulse response is truncated.

Bilinear transform

A nonlinear mapping technique used to design $IIR$ filters by transforming the imaginary axis of the $s$-plane onto the unit circle of the $z$-plane to avoid aliasing.

Frequency warping

The nonlinear relationship between continuous-time frequency \text{\omega} and discrete-time frequency \text{\Omega} inherent in the bilinear transform, given by \text{\Omega} = 2\text{\arctan}(\frac{\text{\omega}T}{2}).

Overlap-add method

A procedure for constructing the filtered output of a long signal by segmenting it into sections, convolving each with the impulse response, and adding the overlapping parts.