Let’s go back to this problem: We take N samples of a sinusoid (or a complex exponential) and we want to estimate its amplitude and frequency by the FFT.

Slides:

Advertisements

Similar presentations

DCSP-12 Jianfeng Feng

Advertisements

DCSP-13 Jianfeng Feng Department of Computer Science Warwick Univ., UK

Department of Kinesiology and Applied Physiology Spectrum Estimation W. Rose

Digital Kommunikationselektronik TNE027 Lecture 5 1 Fourier Transforms Discrete Fourier Transform (DFT) Algorithms Fast Fourier Transform (FFT) Algorithms.

10.1 fourier analysis of signals using the DFT 10.2 DFT analysis of sinusoidal signals 10.3 the time-dependent fourier transform Chapter 10 fourier analysis.

1 Chapter 16 Fourier Analysis with MATLAB Fourier analysis is the process of representing a function in terms of sinusoidal components. It is widely employed.

DFT/FFT and Wavelets ● Additive Synthesis demonstration (wave addition) ● Standard Definitions ● Computing the DFT and FFT ● Sine and cosine wave multiplication.

Windowing Purpose: process pieces of a signal and minimize impact to the frequency domain Using a window – First Create the window: Use the window formula.

Fourier series With coefficients:. Complex Fourier series Fourier transform (transforms series from time to frequency domain) Discrete Fourier transform.

Filtering Filtering is one of the most widely used complex signal processing operations The system implementing this operation is called a filter A filter.

DREAM PLAN IDEA IMPLEMENTATION Introduction to Image Processing Dr. Kourosh Kiani

1 Speech Parametrisation Compact encoding of information in speech Accentuates important info –Attempts to eliminate irrelevant information Accentuates.

Lecture 17 spectral analysis and power spectra. Part 1 What does a filter do to the spectrum of a time series?

Course 2007-Supplement Part 11 NTSC (1) NTSC: 2:1 interlaced, 525 lines per frame, 60 fields per second, and 4:3 aspect ratio horizontal sweep frequency,

Short Time Fourier Transform (STFT)

Discrete Fourier Transform(2) Prof. Siripong Potisuk.

FILTERING GG313 Lecture 27 December 1, A FILTER is a device or function that allows certain material to pass through it while not others. In electronics.

Continuous Time Signals All signals in nature are in continuous time.

Spectral Analysis Spectral analysis is concerned with the determination of the energy or power spectrum of a continuous-time signal It is assumed that.

Discrete Time Periodic Signals A discrete time signal x[n] is periodic with period N if and only if for all n. Definition: Meaning: a periodic signal keeps.

AGC DSP AGC DSP Professor A G Constantinides 1 Digital Filter Specifications Only the magnitude approximation problem Four basic types of ideal filters.

Systems: Definition Filter

Goals For This Class Quickly review of the main results from last class Convolution and Cross-correlation Discrete Fourier Analysis: Important Considerations.

Frequency Domain Representation of Sinusoids: Continuous Time Consider a sinusoid in continuous time: Frequency Domain Representation: magnitude phase.

DTFT And Fourier Transform

GCT731 Fall 2014 Topics in Music Technology - Music Information Retrieval Overview of MIR Systems Audio and Music Representations (Part 1) 1.

Chapter 4 The Frequency Domain of Signals and Systems.

Outline  Fourier transforms (FT)  Forward and inverse  Discrete (DFT)  Fourier series  Properties of FT:  Symmetry and reciprocity  Scaling in time.

Ping Zhang, Zhen Li,Jianmin Zhou, Quan Chen, Bangsen Tian

1 BIEN425 – Lecture 10 By the end of the lecture, you should be able to: –Describe the reason and remedy of DFT leakage –Design and implement FIR filters.

Wavelet transform Wavelet transform is a relatively new concept (about 10 more years old) First of all, why do we need a transform, or what is a transform.

Professor A G Constantinides 1 The Fourier Transform & the DFT Fourier transform Take N samples of from 0 to.(N-1)T Can be estimated from these? Estimate.

Real time DSP Professors: Eng. Julian S. Bruno Eng. Jerónimo F. Atencio Sr. Lucio Martinez Garbino.

1 Spectrum Estimation Dr. Hassanpour Payam Masoumi Mariam Zabihi Advanced Digital Signal Processing Seminar Department of Electronic Engineering Noushirvani.

Environmental and Exploration Geophysics II

Time Frequency Analysis

Lecture#10 Spectrum Estimation

GG313 Lecture 24 11/17/05 Power Spectrum, Phase Spectrum, and Aliasing.

1 Digital Signal Processing Digital Signal Processing  IIR digital filter structures  Filter design.

Summary of Widowed Fourier Series Method for Calculating FIR Filter Coefficients Step 1: Specify ‘ideal’ or desired frequency response of filter Step 2:

Lecture 21: Fourier Analysis Using the Discrete Fourier Transform Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan.

DOPPLER SPECTRA OF WEATHER SIGNALS (Chapter 5; examples from chapter 9)

Fourier Analysis Using the DFT Quote of the Day On two occasions I have been asked, “Pray, Mr. Babbage, if you put into the machine wrong figures, will.

A Brief Journey into Parallel Transmit Jason Su. Description Goal: expose myself to some of the basic techniques of pTx –Replicate in-class results –Explore.

The IIR FILTERs These are highly sensitive to coefficients,

Power Spectral Estimation

Environmental and Exploration Geophysics II tom.h.wilson Department of Geology and Geography West Virginia University Morgantown,

Complex Form of Fourier Series For a real periodic function f(t) with period T, fundamental frequency where is the “complex amplitude spectrum”.

Convergence of Fourier series It is known that a periodic signal x(t) has a Fourier series representation if it satisfies the following Dirichlet conditions:

Professor A G Constantinides 1 Digital Filter Specifications We discuss in this course only the magnitude approximation problem There are four basic types.

Lecture 19 Spectrogram: Spectral Analysis via DFT & DTFT

Fourier series With coefficients:.

Spectral Analysis Spectral analysis is concerned with the determination of the energy or power spectrum of a continuous-time signal It is assumed that.

MECH 373 Instrumentation and Measurements

FFT-based filtering and the

Lab 2: Simple Harmonic Oscillators – Resonance & Damped Oscillations

Fourier Analysis of Signals Using DFT

LINEAR-PHASE FIR FILTERS DESIGN

Sinusoids: continuous time

Ideal Filters One of the reasons why we design a filter is to remove disturbances Filter SIGNAL NOISE We discriminate between signal and noise in terms.

Lecture 16a FIR Filter Design via Windowing

FFTs, Windows, and Circularity

Wavelet transform Wavelet transform is a relatively new concept (about 10 more years old) First of all, why do we need a transform, or what is a transform.

Lecture 18 DFS: Discrete Fourier Series, and Windowing

Finite Wordlength Effects

APPLICATION of the DFT: Estimation of Frequency Spectrum

Clicks and recording.

Lecture 5A: Operations on the Spectrum

Lec.6:Discrete Fourier Transform and Signal Spectrum

ENEE222 Elements of Discrete Signal Analysis Lab 9 1.

Presentation transcript:

Let’s go back to this problem: We take N samples of a sinusoid (or a complex exponential) and we want to estimate its amplitude and frequency by the FFT. What do we get? Estimate the Frequency Spectrum

Take the FFT … FFT Best Estimates based on FFT: Frequency: Amplitude: How good is this estimate?

… again recall what we did… Take a complex exponential of finite length: then its DFT looks like this where we define This is important to understand how good the spectral estimate is.

See the plot of W N / N Main Lobe Side Lobes

See the plot of W N / N in dB’s Main Lobe Side Lobes dB

… and zoom around the main lobe N=64N=256N=1024

Main Lobe The width of the Main Lobe decreases as the data length N increases

Side Lobes Sidelobes are artifacts which don’t belong to the signal. As the data length N increases, the height of the sidelobes stays the same; the height of the first sidelobe is 13dB’s below the maximum

Effect on Frequency Resolution Why all this is important? 1. It has an effect on the frequency resolution. Suppose you have a signal with two frequencies and you take the DFT. See the mainlobes: you can resolve them (2 peaks) you cannot resolve them (1 peak)

Example Consider the signal

… zoom in Consider the signal

Another Example Consider the signal Only One Peak: Cannot Resolve the two frequencies!!!

… take more data points … … of the same signal Two Peaks: Can Resolve the two frequencies.

… zoom in Consider the signal

Now the Sidelobes Consider the signal These are all sidelobes!!!

… add a low power component Consider the signal Because of sidelobes, cannot see the low power frequency component.

Why we have sidelobes? There reason why there are high frequency artifacts (ie sidelobes) is because there is a sharp transition at the edges of the time interval. Remember that the signal is just one period of a periodic signal: One Period Discontinuity!!!

Remedy: use a “window” A remedy is to smooth a signal to “zero” at the edges by multiplying with a window data hamming window windowed data

Use Hamming Window Take the FFT of the “windowed data”: dB k

Use Hamming Window … zoom in 1217 dB k Estimate two frequencies