Signalwaveform, sequnce of
measurements or observations
Processinganalyze, modify, filter,
synthesize
Examples of digital signals
sampled speech waveform
"pixelized" image
Dow-Jones Index
Dsp applications
Filtering (noise reduction)
Pattern recognition (speech, faces,
fingerprints)
Compression
A major difficulty
In many (perhaps most) DSP applications we
don't have complete or perfect knowledge of the signals we wishto process. We are faced with many
unknowns and
uncertainties .
Examples
noisy measurements
unknown signal parameters
noisy system or environmental conditions
natural variability in the signals encountered
Functional magnetic resonance imaging
Challenges are measurement noise and intrinsic
uncertainties in signal behavior.
How can we design signal processing algorithms
in the face of such uncertainty?
Can we model the uncertainty and incorporate
this model into the design process?
Statistical signal processing is the
study of these questions.
Modeling uncertainty
The most widely accepted and commonly used
approach to modeling uncertainty is
Probability
Theory (although other alternatives exist such as Fuzzy
Logic).
Probability Theory models uncertainty by
specifying the
chance of observing certain
signals.
Alternatively, one can view probability as
specifying the degree to which we
believe a
signal reflects the true
state of nature .
Examples of probabilistic models
errors in a measurement (due to an imprecise measuring
device) modeled as realizations of a Gaussian randomvariable.
uncertainty in the phase of a sinusoidal signal
modeled as a uniform random variable on
.
uncertainty in the number of photons stiking a CCD per
unit time modeled as a Poisson random variable.
Statistical inference
A
statistic is a function of
observed data.
Suppose we observe
scalar values
. The following are statistics:
(sample mean)
(the data itself)
(order statistic)
(
,
)
A statistic
cannot depend on
unknown parameters .
Probability is used to model
uncertainty.
Statistics are used to draw
conclusions about probability models.
Probability models our uncertainty about
signals we
may observe.
Statistics reasons from the measured signal to
the population of possible signals.
Statistical signal processing
Step 1
Postulate a probability model (or models) that reasonably
capture the uncertainties at hand
Step 2
Collect data
Step 3
Formulate statistics that allow us to interpret or
understand our probability model(s)
In this class
The two major kinds of problems that we will
study are
detection and
estimation . Most SSP problems fall under one of
these two headings.
Detection theory
Given two (or more) probability models, which
on best explains the signal?
Examples
Decode wireless comm signal into string of 0's and
1's
Pattern recognition
voice recognition
face recognition
handwritten character recognition
Anomaly detection
radar, sonar
irregular, heartbeat
gamma-ray burst in deep space
Estimation theory
If our probability model has free parameters,
what are the best parameter settings to describe the signalwe've observed?
Examples
Noise reduction
Determine parameters of a sinusoid (phase, amplitude,
frequency)
Adaptive filtering
track trajectories of space-craft
automatic control systems
channel equalization
Determine location of a submarine (sonar)
Seismology: estimate depth below ground of an oil
deposit
Detection example
Suppose we observe
tosses of an unfair coin. We
would like to decide which side the coin favors, heads or tails.
Step 1
Assume each toss is a realization of a Bernoulli random
variable.
Must decide
vs.
.
Step 2
Collect data
Step 3
Formulate a useful statistic
If
, guess
. If
, guess
.
Estimation example
Suppose we take
measurements of a DC voltage
with a noisy voltmeter. We would
like to estimate
.
Step 1
Assume a Gaussian noise model
where
.
Step 2
Gather data
Step 3
Compute the sample mean,
and use this as an estimate.
In these examples (
and
), we solved detection and
estimation problems using intuition and heuristics (in Step 3).
This course will focus on developing principled
and mathematically rigorous approaches to detection and estimation,using the theoretical framework of probability and statistics.
Summary
DSPprocessing signals with computer
algorithms.
SSPstatistical DSPprocessing
in the presence of uncertainties and unknowns.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
