Signal processing formula sheet

Basics

Trig Identities

Basic Functions

Sifting Property

System Properties

• Linear:
• Time-Invariant:
• Causal: Output does not depend on future input. For an LTI system, for or for .
• BIBO Stable: . For an LTI system:
  • Continuous-Time:
  • Discrete-Time:

Convolution

• Continuous-Time:
• Discrete-Time:

Differential & Difference Equations

• Differential (CT):
  • System Function:
• Difference (DT): , with
  • System Function:

Common Functions

Definitions

• Inner product
  • Periodic signal:
  • Nonperiodic signal:
  • Inner product with self:

Fourier Transform (FT)

FT Pairs

Time Domain	Frequency Domain

FT Properties

Property	Transform
Linearity
Time Shift
Frequency Shift
Scaling
Differentiation in Time
Differentiation in Frequency
Integration
Convolution-Multiplication
Multiplication-Convolution
Parseval's Theorem

Fourier Transform of Periodic Signals

Periodic Time Domain Signal	Fourier Transform X(f)

Fourier Series (FS)

Basis function:

FS Pairs

Time Domain	Frequency Domain

FS Properties

Property	Transform
Linearity
Time Shift
Frequency Shift
Scaling	(with )
Differentiation in Time
Multiplication-Convolution
Parseval's Theorem

Discrete-Time Fourier Transform (DTFT)

DTFT Pairs

Time Domain	Frequency Domain

DTFT Properties

Property	Transform
Linearity
Time Shift
Frequency Shift
Differentiation in Frequency
Convolution-Multiplication
Multiplication-Convolution
Parseval's Theorem

Discrete-Time Fourier Transform of Periodic signals

Periodic Time Domain Signal	Discrete-Time Fourier Transform

Sampling Theorem

Laplace Transform

Z-Transform

Hilbert Transform

Definition

The Hilbert Transform of is denoted or .

Time domain

Frequency domain

This applies a -90° phase shift to all positive frequency components and a +90° phase shift to all negative frequency components.

Properties

• Linearity:
• Double Transform:
• Orthogonality:
• Energy Preservation:
• Bedrosian Theorem:
  • is a lowpass signal and is a bandpass signal. Their bands don't overlap.
• Common Pairs:
• Analytic Signal

Amplitude Modulation (AM)

Double-Sideband, Suppressed Carrier AM (DSB-SC)

Modulation

Demodulation

Power

Double-Sideband, Transmitted Carrier AM (DSB-TC / Conventional AM)

Modulation

Demodulation

is rectified signal. It brings the band centered from and back to and create harmonics.

• Half-wave rectification:

  Create harmonics around

• Full-wave rectification:

  Create harmonics around

Power

Single-Sideband AM (SSB)

Modulation

• Minus sign () results in the Upper Sideband (USB) signal.
• Plus sign () results in the Lower Sideband (LSB) signal.

Demodulation

Vestigial-Sideband AM (VSB)