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Introduction to Digital Signal Processing

Module by: Mark A. Davenport. E-mail the author

Information, Signals and Systems

Signal processing concerns primarily with signals and systems that operate on signals to extract useful information. In this course our concept of a “signal” will be very broad, encompassing virtually any data that can be represented as an organized “collection” of data.

Example 1

  • A continuous function f(t)f(t)
  • A sequence of discrete data points f[n]f[n]
  • A multi-dimensional array of data
  • Audio, images, video, voltage of antenna
  • Stock prices, potassium concentration in a neuron

Our concept of a “system” will be a black box that takes a signal as input and provides another signal as output.

Example 2

  • Analog-to-digital converters (ADCs)
  • Filters
  • Decimators/Interpolators
  • Matched filters
  • Face recognition systems

In this course we will approach signal processing from the point of view that signals are vectors living in an appropriate vector space, and systems are operators that map signal from one vector space to another. This allows us to use a common mathematical framework to talk about how to:

  • represent signals
  • measure similarity/distance between signals
  • transform signals from one representation to another
  • understand the operation of linear systems on the signals

Since the ficus of this course in on digital signal processing, this will also allow us to use tools from linear algebra to facilitate this understanding.

Digital Signal Processing

DSP is often presented as an alternative to analog signal processing, i.e., instead of a purely analog system as in Figure 1, we can build a digital implementation of an analog system as in Figure 2. This can be advantageous since high-precision analog components are expensive (even compared to the cost of an ADC/DAC).

Figure 1: An analog system.
A purely analog system H that maps a continuous-time signal f(t) to a continuous-time signal g(t)
Figure 2: A digital implementation of an analog system.
A mixed analog-digital system that first samples a continuous-time signal f(t) into a discrete-time signal f[n] which is then processed by a digital system H to obtain the discrete-time signal g[n].  The discrete-time signal g[n] is then input to a digital-to-analog converter to obtain the continuous-time signal g(t).

However, the success of DSP derives to a much greater extent from the facts that:

  1. Discrete-valued signals can be more robust to noise, as illustrated in Figure 3. In Figure 3(a), noise may be impossible to eliminate, but in Figure 3(b) noise can be eliminated entirely by exploiting the discrete structure of the signal.
  2. Once we have a digital, discrete-time signal, we can store it in memory and perform highly complex processing.
Figure 3: (a) An analog signal corrupted with noise; (b) A discrete-valued signal corrupted with noise.
An analog-valued continuous-time signal corrupted by noise.
A discrete-valued continuous-time signal corrupted by noise.

In this course we will consider signal processing systems beyond simple LTI filters. Themes of the course include:

  • Signals as vectors, vector space geometry
  • Signal representations and bases
  • Linear systems analysis and linear algebra
  • “Optimality” in signal processing (e.g., optimal filter design)

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