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

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

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)

Periodic Time Domain Signal	Fourier Transform

Fourier Series (FS)

Basis function:

FS Pairs

Time Domain	Frequency Domain

Time Domain	Frequency Domain

FS Properties

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

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)