- Digital Signal Processing Tutorial
- DSP - Home
- DSP - Signals-Definition
- DSP - Basic CT Signals
- DSP - Basic DT Signals
- DSP - Classification of CT Signals
- DSP - Classification of DT Signals
- DSP - Miscellaneous Signals
- Operations on Signals
- Operations Signals - Shifting
- Operations Signals - Scaling
- Operations Signals - Reversal
- Operations Signals - Differentiation
- Operations Signals - Integration
- Operations Signals - Convolution
- Basic System Properties
- DSP - Static Systems
- DSP - Dynamic Systems
- DSP - Causal Systems
- DSP - Non-Causal Systems
- DSP - Anti-Causal Systems
- DSP - Linear Systems
- DSP - Non-Linear Systems
- DSP - Time-Invariant Systems
- DSP - Time-Variant Systems
- DSP - Stable Systems
- DSP - Unstable Systems
- DSP - Solved Examples
- Z-Transform
- Z-Transform - Introduction
- Z-Transform - Properties
- Z-Transform - Existence
- Z-Transform - Inverse
- Z-Transform - Solved Examples
- Discrete Fourier Transform
- DFT - Introduction
- DFT - Time Frequency Transform
- DTF - Circular Convolution
- DFT - Linear Filtering
- DFT - Sectional Convolution
- DFT - Discrete Cosine Transform
- DFT - Solved Examples
- Fast Fourier Transform
- DSP - Fast Fourier Transform
- DSP - In-Place Computation
- DSP - Computer Aided Design
- Digital Signal Processing Resources
- DSP - Quick Guide
- DSP - Useful Resources
- DSP - Discussion
DSP - DFT Linear Filtering
DFT provides an alternative approach to time domain convolution. It can be used to perform linear filtering in frequency domain.
Thus,$Y(\omega) = X(\omega).H(\omega)\longleftrightarrow y(n)$.
The problem in this frequency domain approach is that $Y(\omega)$, $X(\omega)$ and $H(\omega)$ are continuous function of ω, which is not fruitful for digital computation on computers. However, DFT provides sampled version of these waveforms to solve the purpose.
The advantage is that, having knowledge of faster DFT techniques likes of FFT, a computationally higher efficient algorithm can be developed for digital computer computation in comparison with time domain approach.
Consider a finite duration sequence, $[x(n) = 0,\quad for,n<0\quad and\quad n\geq L]$ (generalized equation), excites a linear filter with impulse response $[h(n) = 0,\quad forn<0\quad and\quad n\geq M]$.
$$x(n)y(n)$$ $$output = y(n) = \sum_{k = 0}^{M-1}h(k).x(n-k)$$From the convolution analysis, it is clear that, the duration of y(n) is L+M−1.
In frequency domain,
$$Y(\omega) = X(\omega).H(\omega)$$Now, $Y(\omega)$ is a continuous function of ω and it is sampled at a set of discrete frequencies with number of distinct samples which must be equal to or exceeds $L+M-1$.
$$DFT\quad size = N\geq L+M-1$$With $\omega = \frac{2\pi}{N}k$,
$Y(\omega) = X(k).H(k)$, where k=0,1,….,N-1
Where, X(k) and H(k) are N-point DFTs of x(n) and h(n) respectively. $x(n)\& h(n)$ are padded with zeros up to the length N. It will not distort the continuous spectra $X(\omega)$ and $H(\omega)$. Since $N\geq L+M-1$, N-point DFT of output sequence y(n) is sufficient to represent y(n) in frequency domain and these facts infer that the multiplication of N-point DFTs of X(k) and H(k), followed by the computation of N-point IDFT must yield y(n).
This implies, N-point circular convolution of x(n) and H(n) with zero padding, equals to linear convolution of x(n) and h(n).
Thus, DFT can be used for linear filtering.
Caution − N should always be greater than or equal to $L+M-1$. Otherwise, aliasing effect would corrupt the output sequence.