Dr. Nikos Desypris, ndessipris@yahoo.com Oct. 2017 Lecture 3 Digital Television Dr. Nikos Desypris, ndessipris@yahoo.com Oct. 2017 Lecture 3

Course Outline Signals and frequencies (harmonics) Fourier Transform FT & Power Spectrum Nyquist theorem Discrete FT (DFT) and Inverse DFT (IDFT) Discrete Cosine Transform (DCT) and Inverse DCT (IDCT)

What is a signal? In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity. .... In the physical world, any quantity measurable through time or over space can be taken as a signal. .... Despite the complexity of such systems, their outputs and inputs can often be represented as simple quantities measurable through time or across space (www.wikipedia.org).

What is a signal? (cnt.) Signals are generally analogue, meaning that they have values at any time or at any spatial location. Often signals are expressed with the notion of function i.e.: a signal is a function of time or a function of length.

Fourier Analysis of a Periodic Signal

Euler Formula of a sinewave To reconstruct a sin/cos of period T=1/f, I need to know: A&φ

Fourier Transform

Time & Frequency domain Continuous signal

Time & Frequency domain Discrete signal

Time & Frequency domain Discrete signal

Time & Frequency domain Discrete signal

Continuous/Finite/Digital signals Continuous signals are signals that theoretically have an infinite duration (over time or across space or both). Finite signals are signals that have finite duration, they are either restricted in a specific period of time (i.e from t0 to t0+Δt) or have a specific spatial extend (i.e. a line that extend from location x0 to location x0+Δx). A finite signal is an analogue signal.  A digital signal is an analogue signal that has been sampled at (usually) repetitive intervals. Digital signals are approximations of analogue signals and exist only for the points / locations / intervals where they were sampled.

Fourier Transform in 1-D The theory of Fourier Transform tells us that any signal can be expressed as a sum of a series of sinusoids of different frequencies. Practically this means that we have two representations of a signal: one representation in the time domain where the signal is a function of time and - the other representation in which the signal is a representation of frequencies (i.e no time is involved).

Fourier Transform in 1-D (cnt.) The important point to notice is that both representations depict the same signal and we can move from time domain to frequency domain through mathematical transforms. Since the transforms are rather complicated we will try to present them in a schematic way in order to grasp the basic notions first in one dimension and secondly in two dimensions.

Fourier Transform of signals FT of aperiodic signals (infinite number of frequencies, infinite range) Fourier Transform can work on Aperiodic Signals. Fourier Transform is an infinite sum of infinitesimal sinusoids. Fourier Transform has an inverse transform, that allows for conversion from the frequency domain back to the time domain FT of finite periodic signals (infinite no of frequencies, specific range) FT of a periodic bandpass signal (specific range of frequencies and specific range/length)

Fourier Transform of signals Time Domain Frequency Domain Aperiodical Continuous, Infinite Periodical Continuous, Finite Continuous, Infinite Digital Signals Discrete, Finite (periodical)

FT & Inverse FT in one dimension

FT & Inverse FT in one dimension

DFT & Inverse DFT in 1D The DFT of a periodical signal of one discrete variable extended over N samples is (Discrete Fourier Transform DFT){f(x)} = Given F(u), f(t) can be obtained through the Inverse Discrete Fourier Transform - IDFT (Inverse DFT){F(u)} =

The Nyquist sampling theorem Recorded image Sensor 24 elements per image width 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Recorded image Sensor 24 elements per image width 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Recorded image Sensor: 6 elements per image width 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Recorded image Sensor: 6 elements per image width 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Sensor size? 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Recorded image Sensor: 16 elements per image size 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Recorded image Sensor 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem The Nyquist theorem: In order not to lose information, we must sample the analogue signal with a rate at least double the maximum frequency component that exists in the signal The only solution to the aliasing problem is to ensure that the sampling rate is higher than twice of the highest frequency present in the signal. If that is not possible, then use an anti-aliasing filter to screen out those frequencies higher than ½ of the sampling frequency before sampling, assuming the removed frequencies are not of importance.

The Nyquist sampling theorem In practice we need to sample the image above the Nyquist rate! Sensor 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem Sampling points Original signal Sampled signal Two different sinusoids that fit the same set of samples; better to go above Nyquist rate.

The Nyquist sampling theorem Recorded image ?? Sensor size 16 2d Image of a sinus Freq = 8 cycles per image width White -> 256 Profile of an image line Black -> 0

The Nyquist sampling theorem White -> 256 Profile of an image line Black -> 0 White -> 256 Profile of an image line Black -> 0

Fourier Transform of signals FT of aperiodic signals (infinite number of frequencies, infinite range) Fourier Transform can work on Aperiodic Signals. Fourier Transform is an infinite sum of infinitesimal sinusoids. Fourier Transform has an inverse transform, that allows for conversion from the frequency domain back to the time domain FT of finite periodic signals (infinite no of frequencies, specific range) FT of a periodic bandpass signal (specific range of frequencies and specific range/length)

Fourier Transform of signals Time Domain Frequency Domain Aperiodical Continuous, Infinite Periodical Continuous, Finite Continuous, Infinite Digital Signals Discrete, Finite (periodical)

DFT & Inverse DFT in 1D The DFT of a periodical signal of one discrete variable extended over N samples is (Discrete Fourier Transform DFT){f(x)} = Given F(u), f(t) can be obtained through the Inverse Discrete Fourier Transform - IDFT (Inverse DFT){F(u)} =

1D DFT by example

1D DFT by example

1D DFT by example Spatial Domain f(0), f(1), f(2), f(3) Frequency Domain F(0), F(1), F(2), F(3) F(u) = R(u) + j I(u) F(u) = |F(u)| * exp (-j φ(u))

Euler Formula of a sinewave To reconstruct a sin/cos of period T=1/f, I need to know: A&φ

Euler Formula of a sinewave http://treeblurb.com/dev_math/sin_canv00.html To reconstruct a sin/cos of period T=1/f, I need to know: A&φ

1D DFT by example Spatial Domain f(0), f(1), f(2), f(3) Frequency Domain F(0), F(1), F(2), F(3) F(u) = R(u) + j I(u) F(u) = |F(u)| * exp (-j φ(u)) So we have a sin/cos of amplitude F(u) and phase φ(u) For the time being we ignore φ(u) and we focus on |F(u)| = sqrt (R*R + I*I) called the Power Spectrum, showing the magnitude of each frequency

1D DFT by example F(0) is the mean value (called also DC component) F(1) is the basic frequency (one cycle through 4 points); F(2) is the –ve of F(1) F(2) corresponds to the 1st harmonic (twice the basic frequency) and is double the frequency of F1

1D DFT by example

From 1D DFT -> 2D DFT The values of f(0), f(1), f(2), and f(3) are now considered pixels of a line of an image f(0) f(1) f(2) f(3) So how we perform a 2D DFT?

2D DFT of a sine wave 2D sinewave 8 horizontal cycles per 256 pixels

2D DFT of a sine wave 2D sinewave 32 cycles per 256 pixels

2D DFT of a sine wave 2D sinewave 8 vertical cycles per 256 pixels

2D DFT of a sine wave 2D sinewave 32 vertical cycles per 256 pixels

2D DFT of a sine wave 2D sinewave 16 diagonal cycles per 256 pixels (width/height)

2D DFT of a sine wave 2D sinewave 16,2 diagonal cycles per 256 pixels (width/height)

2D DFT of a sine wave The figure below, presents replication of the sinusoid with 16.2 diagonal cycles. Note the discrepancies at the edges that result in high frequencies presence in the Power Spectrum

Images and their DFTs Power spectrum of a forest

Images and their DFTs Coins and the corresponding power spectrum of coins. Please note the frequencies multiples of the coin size.

Images and their DFTs Falling stars and its power Spectrum. Please note the existence of high frequencies perpendicular to the direction of the falling stars.

LPF in frequency domain Low pass filtering (step edge causing ringing effect)

Special filter Filter reduces low frequencies (by 0,5) & accentuates high frequencies (by 4.0)

DCT

DCT

DCT

DCT

DCT

DCT

DCT

DCT

DCT