- Signals & Systems Home
- Signals & Systems Overview
- Introduction
- Signals Basic Types
- Signals Classification
- Signals Basic Operations
- Systems Classification
- Types of Signals
- Representation of a Discrete Time Signal
- Continuous-Time Vs Discrete-Time Sinusoidal Signal
- Even and Odd Signals
- Properties of Even and Odd Signals
- Periodic and Aperiodic Signals
- Unit Step Signal
- Unit Ramp Signal
- Unit Parabolic Signal
- Energy Spectral Density
- Unit Impulse Signal
- Power Spectral Density
- Properties of Discrete Time Unit Impulse Signal
- Real and Complex Exponential Signals
- Addition and Subtraction of Signals
- Amplitude Scaling of Signals
- Multiplication of Signals
- Time Scaling of Signals
- Time Shifting Operation on Signals
- Time Reversal Operation on Signals
- Even and Odd Components of a Signal
- Energy and Power Signals
- Power of an Energy Signal over Infinite Time
- Energy of a Power Signal over Infinite Time
- Causal, Non-Causal, and Anti-Causal Signals
- Rectangular, Triangular, Signum, Sinc, and Gaussian Functions
- Signals Analysis
- Types of Systems
- What is a Linear System?
- Time Variant and Time-Invariant Systems
- Linear and Non-Linear Systems
- Static and Dynamic System
- Causal and Non-Causal System
- Stable and Unstable System
- Invertible and Non-Invertible Systems
- Linear Time-Invariant Systems
- Transfer Function of LTI System
- Properties of LTI Systems
- Response of LTI System
- Fourier Series
- Fourier Series
- Fourier Series Representation of Periodic Signals
- Fourier Series Types
- Trigonometric Fourier Series Coefficients
- Exponential Fourier Series Coefficients
- Complex Exponential Fourier Series
- Relation between Trigonometric & Exponential Fourier Series
- Fourier Series Properties
- Properties of Continuous-Time Fourier Series
- Time Differentiation and Integration Properties of Continuous-Time Fourier Series
- Time Shifting, Time Reversal, and Time Scaling Properties of Continuous-Time Fourier Series
- Linearity and Conjugation Property of Continuous-Time Fourier Series
- Multiplication or Modulation Property of Continuous-Time Fourier Series
- Convolution Property of Continuous-Time Fourier Series
- Convolution Property of Fourier Transform
- Parseval’s Theorem in Continuous Time Fourier Series
- Average Power Calculations of Periodic Functions Using Fourier Series
- GIBBS Phenomenon for Fourier Series
- Fourier Cosine Series
- Trigonometric Fourier Series
- Derivation of Fourier Transform from Fourier Series
- Difference between Fourier Series and Fourier Transform
- Wave Symmetry
- Even Symmetry
- Odd Symmetry
- Half Wave Symmetry
- Quarter Wave Symmetry
- Wave Symmetry
- Fourier Transforms
- Fourier Transforms Properties
- Fourier Transform – Representation and Condition for Existence
- Properties of Continuous-Time Fourier Transform
- Table of Fourier Transform Pairs
- Linearity and Frequency Shifting Property of Fourier Transform
- Modulation Property of Fourier Transform
- Time-Shifting Property of Fourier Transform
- Time-Reversal Property of Fourier Transform
- Time Scaling Property of Fourier Transform
- Time Differentiation Property of Fourier Transform
- Time Integration Property of Fourier Transform
- Frequency Derivative Property of Fourier Transform
- Parseval’s Theorem & Parseval’s Identity of Fourier Transform
- Fourier Transform of Complex and Real Functions
- Fourier Transform of a Gaussian Signal
- Fourier Transform of a Triangular Pulse
- Fourier Transform of Rectangular Function
- Fourier Transform of Signum Function
- Fourier Transform of Unit Impulse Function
- Fourier Transform of Unit Step Function
- Fourier Transform of Single-Sided Real Exponential Functions
- Fourier Transform of Two-Sided Real Exponential Functions
- Fourier Transform of the Sine and Cosine Functions
- Fourier Transform of Periodic Signals
- Conjugation and Autocorrelation Property of Fourier Transform
- Duality Property of Fourier Transform
- Analysis of LTI System with Fourier Transform
- Relation between Discrete-Time Fourier Transform and Z Transform
- Convolution and Correlation
- Convolution in Signals and Systems
- Convolution and Correlation
- Correlation in Signals and Systems
- System Bandwidth vs Signal Bandwidth
- Time Convolution Theorem
- Frequency Convolution Theorem
- Energy Spectral Density and Autocorrelation Function
- Autocorrelation Function of a Signal
- Cross Correlation Function and its Properties
- Detection of Periodic Signals in the Presence of Noise (by Autocorrelation)
- Detection of Periodic Signals in the Presence of Noise (by Cross-Correlation)
- Autocorrelation Function and its Properties
- PSD and Autocorrelation Function
- Sampling
- Signals Sampling Theorem
- Nyquist Rate and Nyquist Interval
- Signals Sampling Techniques
- Effects of Undersampling (Aliasing) and Anti Aliasing Filter
- Different Types of Sampling Techniques
- Laplace Transform
- Laplace Transforms
- Common Laplace Transform Pairs
- Laplace Transform of Unit Impulse Function and Unit Step Function
- Laplace Transform of Sine and Cosine Functions
- Laplace Transform of Real Exponential and Complex Exponential Functions
- Laplace Transform of Ramp Function and Parabolic Function
- Laplace Transform of Damped Sine and Cosine Functions
- Laplace Transform of Damped Hyperbolic Sine and Cosine Functions
- Laplace Transform of Periodic Functions
- Laplace Transform of Rectifier Function
- Laplace Transforms Properties
- Linearity Property of Laplace Transform
- Time Shifting Property of Laplace Transform
- Time Scaling and Frequency Shifting Properties of Laplace Transform
- Time Differentiation Property of Laplace Transform
- Time Integration Property of Laplace Transform
- Time Convolution and Multiplication Properties of Laplace Transform
- Initial Value Theorem of Laplace Transform
- Final Value Theorem of Laplace Transform
- Parseval's Theorem for Laplace Transform
- Laplace Transform and Region of Convergence for right sided and left sided signals
- Laplace Transform and Region of Convergence of Two Sided and Finite Duration Signals
- Circuit Analysis with Laplace Transform
- Step Response and Impulse Response of Series RL Circuit using Laplace Transform
- Step Response and Impulse Response of Series RC Circuit using Laplace Transform
- Step Response of Series RLC Circuit using Laplace Transform
- Solving Differential Equations with Laplace Transform
- Difference between Laplace Transform and Fourier Transform
- Difference between Z Transform and Laplace Transform
- Relation between Laplace Transform and Z-Transform
- Relation between Laplace Transform and Fourier Transform
- Laplace Transform – Time Reversal, Conjugation, and Conjugate Symmetry Properties
- Laplace Transform – Differentiation in s Domain
- Laplace Transform – Conditions for Existence, Region of Convergence, Merits & Demerits
- Z Transform
- Z-Transforms (ZT)
- Common Z-Transform Pairs
- Z-Transform of Unit Impulse, Unit Step, and Unit Ramp Functions
- Z-Transform of Sine and Cosine Signals
- Z-Transform of Exponential Functions
- Z-Transforms Properties
- Properties of ROC of the Z-Transform
- Z-Transform and ROC of Finite Duration Sequences
- Conjugation and Accumulation Properties of Z-Transform
- Time Shifting Property of Z Transform
- Time Reversal Property of Z Transform
- Time Expansion Property of Z Transform
- Differentiation in z Domain Property of Z Transform
- Initial Value Theorem of Z-Transform
- Final Value Theorem of Z Transform
- Solution of Difference Equations Using Z Transform
- Long Division Method to Find Inverse Z Transform
- Partial Fraction Expansion Method for Inverse Z-Transform
- What is Inverse Z Transform?
- Inverse Z-Transform by Convolution Method
- Transform Analysis of LTI Systems using Z-Transform
- Convolution Property of Z Transform
- Correlation Property of Z Transform
- Multiplication by Exponential Sequence Property of Z Transform
- Multiplication Property of Z Transform
- Residue Method to Calculate Inverse Z Transform
- System Realization
- Cascade Form Realization of Continuous-Time Systems
- Direct Form-I Realization of Continuous-Time Systems
- Direct Form-II Realization of Continuous-Time Systems
- Parallel Form Realization of Continuous-Time Systems
- Causality and Paley Wiener Criterion for Physical Realization
- Discrete Fourier Transform
- Discrete-Time Fourier Transform
- Properties of Discrete Time Fourier Transform
- Linearity, Periodicity, and Symmetry Properties of Discrete-Time Fourier Transform
- Time Shifting and Frequency Shifting Properties of Discrete Time Fourier Transform
- Inverse Discrete-Time Fourier Transform
- Time Convolution and Frequency Convolution Properties of Discrete-Time Fourier Transform
- Differentiation in Frequency Domain Property of Discrete Time Fourier Transform
- Parseval’s Power Theorem
- Miscellaneous Concepts
- What is Mean Square Error?
- What is Fourier Spectrum?
- Region of Convergence
- Hilbert Transform
- Properties of Hilbert Transform
- Symmetric Impulse Response of Linear-Phase System
- Filter Characteristics of Linear Systems
- Characteristics of an Ideal Filter (LPF, HPF, BPF, and BRF)
- Zero Order Hold and its Transfer Function
- What is Ideal Reconstruction Filter?
- What is the Frequency Response of Discrete Time Systems?
- Basic Elements to Construct the Block Diagram of Continuous Time Systems
- BIBO Stability Criterion
- BIBO Stability of Discrete-Time Systems
- Distortion Less Transmission
- Distortionless Transmission through a System
- Rayleigh’s Energy Theorem
What is the Frequency Response of Discrete-Time Systems?
Frequency Response of Discrete-Time Systems
A spectrum of input sinusoids is applied to a linear time-invariant (LTI) discrete-time system to obtain the frequency response of the system. The frequency response of the discrete-time system gives the magnitude and phase response of the system to the input sinusoids at all frequencies.
Let the impulse response of an LTI discrete-time system be $\mathrm{h(n)}$ and the input to the system be a complex exponential function, i.e., $\mathrm{x(n) \:=\: e^{j \omega n}}$. Then, the output $\mathrm{y(n)}$ of the system is obtained by using the convolution theorem, i.e.,
$$\mathrm{y(n) \:=\: h(n)\:\cdot\: x(n) \:=\: \sum_{k=-\infty}^{\infty}\: h(k) x(n\:-\:k)}$$
As the input to the system is $\mathrm{x(n) \:=\: e^{j \omega n}}$, then:
$$\mathrm{y(n) \:=\: \sum_{k=-\infty}^{\infty}\: h(k)\: e^{j \omega (n-k)}}$$
$$\mathrm{y(n) \:=\: e^{j \omega n}\: \sum_{k=-\infty}^{\infty}\: h(k)\: e^{-j \omega k}}$$
$$\mathrm{\therefore\:y(n) \:=\: e^{j \omega n} \:\cdot\: H(\omega)}$$
Where:
- $\mathrm{e^{j \omega n}}$ is the input sequence, and
- $\mathrm{H(\omega)}$ is the frequency response of the discrete-time system.
Therefore, the output of the discrete-time system is identical to the input, modified in magnitude and phase by $\mathrm{H(\omega)}$. The frequency response $\mathrm{H(\omega)}$ of the discrete-time system is a complex quantity and can be expressed as:
$$\mathrm{H(\omega) \:=\: |H(\omega)| \:e^{j \angle H(\omega)}}$$
Where:
- $\mathrm{|H(\omega)|}$ is called the magnitude response of the discrete-time system, and
- $\mathrm{\angle\: H(\omega)}$ is called the phase response of the system.
Also, the graph plotted between $\mathrm{|H(\omega)|}$ and $\mathrm{\omega}$ is called the magnitude response plot, and the graph plotted between $\mathrm{\angle H(\omega)}$ and $\mathrm{\omega}$ is called the phase response plot.
Properties of Frequency Response of Discrete-Time Systems
If the impulse response $\mathrm{h(n)}$ of the LTI discrete-time system is a real sequence, then the frequency response $\mathrm{H(\omega)}$ possesses the following properties:
- The frequency response $\mathrm{H(\omega)}$ takes on values for all $\mathrm{\omega}$.
- The frequency response $\mathrm{H(\omega)}$ is periodic in $\mathrm{\omega}$ with a time period $\mathrm{2\pi}$.
- $\mathrm{|H(\omega)|}$\, i.e., the magnitude response of the system, is an even function of $\mathrm{\omega}$ and is symmetrical about $\mathrm{\pi}$.
- $\mathrm{\angle H(\omega)}$, i.e., the phase response of the system, is an odd function of $\mathrm{\omega}$ and it is anti-symmetrical about $\mathrm{\pi}$.
Numerical Example
Find the frequency response of the following discrete-time causal system:
$$\mathrm{y(n) \:-\: 2y(n\:-\:1) \:+\: \frac{2}{9}y(n\:-\:2) \:=\: x(n) \:-\: \frac{3}{5}x(n\:-\:1)}$$
Solution
If $\mathrm{X(\omega)}$ and $\mathrm{Y(\omega)}$ are the Fourier transforms of the input and output sequences, then the frequency response of the discrete-time system is given by:
$$\mathrm{H(\omega) \:=\: \frac{Y(\omega)}{X(\omega)}}$$
Now, the equation describing the system is:
$$\mathrm{y(n) \:-\: 2y(n\:-\:1) \:+\: \frac{2}{9}y(n\:-\:2) \:=\: x(n) \:-\: \frac{3}{5}x(n\:-\:1)}$$
Taking the discrete-time Fourier transform on both sides, we get:
$$\mathrm{Y(\omega) \:-\: 2e^{-j \omega} Y(\omega) \:+\: \frac{2}{9}e^{-j 2 \omega} Y(\omega) \:=\: X(\omega) \:-\: \frac{3}{5}e^{-j \omega}\: X(\omega)}$$
$$\mathrm{\Rightarrow\:Y(\omega) \left( 1 \:-\: 2e^{-j \omega} \:+\: \frac{2}{9}e^{-j 2 \omega} \right) \:=\: X(\omega) \left( 1 \:-\: \frac{3}{5}e^{-j \omega} \right)}$$
$$\mathrm{\Rightarrow\:\frac{Y(\omega)}{X(\omega)}\:=\:\frac{\left(1\:-\:\frac{3}{5}e^{-j \omega}\right)}{\left(1\:-\:2e^{-j\omega}\:+\:\frac{2}{9}e^{-j2\omega}\right)}}$$
$$\mathrm{\therefore\:H(\omega) \:=\: \frac{Y(\omega)}{X(\omega)} \:=\: \frac{e^{j \omega}}{\left(e^{j \omega} \:-\: \frac{3}{5}\right)\left(e^{j 2 \omega} \:-\: 2e^{j \omega} \:+\: \frac{2}{9}\right)}}$$