Digital Signal Processing
Digital Signal Processing (DSP) is the mathematical manipulation of discrete-time, discrete-amplitude representations of signals—obtained by sampling and quantising an analogue source—to extract information, filter noise, compress data, or transform the signal into a desired form. Core operations include convolution, Fourier analysis via the FFT, FIR/IIR filtering, and modulation; these are implemented in real time on dedicated DSP chips or general-purpose microprocessors. DSP underpins audio processing, medical imaging, telecommunications, radar, sonar, and machine learning inference at the edge.
Key Formula
Y(z) = H(z) * X(z), where H(z) = sum(b_k * z^-k) / (1 - sum(a_k * z^-k))
LaTeX: Y(z) = H(z) \cdot X(z), \quad H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 - \sum_{k=1}^{N} a_k z^{-k}}
| Symbol | Meaning | Unit |
|---|---|---|
| Y(z) | Z-transform of output signal | dimensionless |
| H(z) | Transfer function of digital filter | dimensionless |
| X(z) | Z-transform of input signal | dimensionless |
| b_k | Feed-forward (numerator) coefficients | dimensionless |
| a_k | Feedback (denominator) coefficients | dimensionless |
| z | Complex frequency variable | dimensionless |
Worked Example
Problem
A first-order IIR low-pass filter has the transfer function H(z) = 0.1 / (1 − 0.9z⁻¹). Given an input sequence x[n] = {1, 0, 0, 0} (unit impulse), find the first four output samples y[n] using the difference equation.
Solution
Step 1: Write the difference equation from H(z). y[n] = 0.1·x[n] + 0.9·y[n−1], with y[−1] = 0. Step 2: Compute iteratively. n=0: y[0] = 0.1×1 + 0.9×0 = 0.1 n=1: y[1] = 0.1×0 + 0.9×0.1 = 0.09 n=2: y[2] = 0.1×0 + 0.9×0.09 = 0.081 n=3: y[3] = 0.1×0 + 0.9×0.081 = 0.0729
Answer
y[0]=0.1, y[1]=0.09, y[2]=0.081, y[3]=0.0729 — an exponentially decaying impulse response.
FIR vs IIR Digital Filter Comparison
| Property | FIR Filter | IIR Filter |
|---|---|---|
| Stability | Always stable | Conditionally stable |
| Phase response | Linear phase possible | Non-linear phase |
| Feedback | No feedback | Feedback required |
| Order for sharp cutoff | High (100s of taps) | Low (10–20 stages) |
| Computational cost | Higher | Lower |
| Analogous to | Moving average | Analogue RLC filter |
Interactive Tools
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Related Terms
Electrical Modulation
Electrical modulation is the process of varying one or more properties of a high-frequency carrier signal—such as amplitude, frequency, or phase—in proportion to a lower-frequency information signal. It is fundamental to all modern communication systems because modulated carriers can travel long distances with minimal energy loss and can be multiplexed to share a single channel. Applications include AM/FM radio, cellular networks, Wi-Fi, and satellite communications.
Microprocessor
A microprocessor is an integrated circuit that incorporates the functions of a central processing unit (CPU) on a single semiconductor chip, executing stored-program instructions to perform arithmetic, logic, control, and input/output operations. Modern microprocessors contain billions of transistors and integrate multiple processing cores, cache hierarchies, floating-point units, and memory controllers to achieve high computational throughput at low power. They are the foundational component of computers, smartphones, embedded controllers, and virtually all programmable digital systems.
Antenna (electrical)
An antenna is a transducer that converts between guided electromagnetic waves in a transmission line and freely propagating electromagnetic waves in space, enabling wireless transmission and reception of signals. Key performance parameters include gain, radiation pattern, bandwidth, polarisation, and impedance; these are governed by the antenna's geometry, size relative to the operating wavelength, and surrounding environment. Antennas are central to radio broadcasting, mobile communications, radar, satellite links, and wireless sensor networks.
From Latin "digitalis" (of or pertaining to the finger, hence finger-counted/discrete) and Latin "signum" (sign, signal). "Processing" from Latin "processus". The field emerged in the late 1960s with the development of the Fast Fourier Transform by Cooley and Tukey (1965) and the proliferation of integrated circuits.