Discrete Time Signals And
Systems
What is signal?
• In signal processing, a signal is a function that
conveys information about a phenomenon.
• In electronics and telecommunications, it refers to any
time varying voltage, current, or electromagnetic wave
that carries information.
• Generally, signal is a time varying physical
phenomenon which is intended to convey information.
OR
Signal is a function of time.
OR
Signal is a function of one or more independent
variables, which contain some information.
Types of Signal
• Signals are classified into the following
categories:
Deterministic and Non-deterministic Signals
Continuous Time and Discrete Time Signals
Even and Odd Signals
Periodic and Aperiodic Signals
Energy and Power Signals
Deterministic and Non-deterministic Signals
• A signal is said to be deterministic if there is no uncertainty with respect to
its value at any instant of time. Or, signals which can be defined exactly by
a mathematical formula are known as deterministic signals.
• A signal is said to be non-deterministic if there is uncertainty with respect
to its value at some instant of time.
• Non-deterministic signals are random in nature hence they are called
random signals. Random signals cannot be described by a mathematical
equation. They are modelled in probabilistic terms.
Continuous Time Signals
• Continuous-time signal is the “function of continuous-time variable
that has uncountable or infinite set of numbers in its sequence”.
• The continuous-time signal can be represented and defined at any
instant of the time in its sequence.
• The examples for continuous-time signals are sine waves, cosine
waves, triangular waves, and so on.
• The electrical signals also behave as continuous-time signals when
these are derived in proportion with the physical parameters such as
pressure, temperature, sound, and so on.
Discrete Time Signals
• Discrete-time signal is the “function of discrete-time variable that
has countable or finite set of numbers in its sequence”.
• It is a digital representation of continuous-time signal.
• The discrete-time signal can be represented and defined at certain
instants of time in its sequence i.e. the discrete-time signal is able to
define only at the sampling instants.
• The output data from a computer is one of the examples of discretetime signals.
What is Digital Signal?
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Digital signal can be obtained from the discrete-time signal by quantizing and
encoding the sample values.
The discrete-time signals are represented with binary bits and stored on the digital
medium.
For example, if a signal is in range 0-5 volts; a discrete time signal (not quantized)
can take values like 0.2, 0.02, 0.0005, 2, 4 etc. (just anything in the range 0-5 volts).
But if it is uniformly quantized with say N quantization levels, then range 0-5 will
be divided into N equal parts, each part will be assigned one value (which is in the
range of that specific part), hence giving a set of N amplitude values. The digital
signal can take any value out of these N values only ( and not just any value).
Representation of discrete signal
• Discrete-time signals are often depicted graphically as follows:
• The value x[n] is undefined for non-integer values of n.
• Sequences can be manipulated in several ways. The sum and product of
two sequences x[n] and y[n] are defined as the sample-by-sample sum and
product respectively.
• Multiplication of x[n] by a is defined as the multiplication of each sample
value by a.
• A sequence y[n] is a delayed or shifted version of x[n] if
y[n]=x[n-n0]
with n0 an integer.
Standard Discrete signals
• The unit sample sequence
is defined as
• This sequence is often referred to as a discrete-time
impulse, or just impulse.
• It plays the same role for discrete-time signals as the
Dirac delta function does for continuous-time signals
Unit Step Sequence
Exponential sequences
Sinusoidal Sequence
Even and odd signals
• A discrete signal is said to be even or symmetric if x[-n] = x[n]
• A discrete signal is said to be odd or asymmetric if x[-n] = −x[n]
Periodic and Aperiodic Signals
• A discrete time signal is periodic if and only if, it satisfies the
following condition
x(n+N)=x(n)
where Ω0 is the frequency
Energy and Power Signals
What is system?
• System is a device or combination of devices, which can operate on
signals and produces corresponding response
• A system is any process that produces an output signal in response
to an input signal.
• Continuous systems input and output continuous signals, such as in
analog electronics.
• Discrete systems input and output discrete signals, such as computer
programs that manipulate the values stored in arrays.
Discrete Time Systems
Memory less systems
• A system is memory-less if the output y[n] depends only on x[n] at
the same n.
• For example, y[n]=(x[n])2 is memory less but the ideal delay
y[n]=x[n-nd] is not unless nd = 0.
Linear systems
Time-Invariant systems
Example of continuous time system
Causal and Non Causal system
Causal Signal
h[k] is the response of a linear system to the impulse input δ[k]
Stability
Linear Time-Invariant (LTI) systems
• If the linearity property is combined with the representation of a general
sequence as a linear combination of delayed impulses, then it follows that
a linear time-invariant (LTI) system can be completely characterised by its
impulse response.
• Suppose hk[n]. is the response of a linear system to the impulse δ[n-k] at
n = k. Since (from slide 12 & 28)
• The principle of superposition means that
• If the system is additionally time invariant, then the response
to δ[n-k] is h[n-k].The previous equation then becomes
• This expression is called the convolution sum. Therefore, a LTI
system has the property that given h[n], we can find y[n] for
any input x[n]
• Alternatively, y[n] is the convolution of x[n] with h[n] denoted
as follows:
y[n]=x[n]*h[n]
Stability of LTI systems
Representation of signal on orthogonal
basis
Difference Equation
A linear constant-coefficient difference equation serves as a way to express
represent the input/output relationship to a given LTI discrete-time system.
From this equation, note that y[n−k] represents the outputs and x[n−k]
represents the inputs.
The value of N represents the order of the difference equation (or order of the
system) and corresponds to the memory of the system being represented.
Because this equation relies on past values of the output, in order to compute
a numerical solution, certain past outputs, referred to as the initial conditions,
must be known.
Solution of LCCDE
Digital Signal Processing (4th Edition) by John G. Proakis, Dimitris K Manolakis,
Pg: 98 onwards, solved problems
What is Digital signal processing?
• Digital signal processing (DSP) is the process of analyzing and
modifying a signal to optimize or improve its efficiency or
performance.
• It involves applying various mathematical and computational
algorithms to analog and digital signals to produce a signal that's of
higher quality than the original signal.
• DSP is primarily used to detect errors, and to filter and compress
analog signals in transit.
• It is a type of signal processing performed through a digital signal
processor or a similarly capable device that can execute DSP
specific processing algorithms.
• Typically, DSP first converts an analog signal into a digital signal
and then applies signal processing techniques and algorithms.
• For example, when performed on audio signals, DSP helps reduce
noise and distortion. Some of the applications of DSP include audio
signal processing, digital image processing, speech recognition,
biomedicine and more.
Block Diagram of DSP
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The first step is to get an electrical signal. The transducer (in this case, a microphone)
converts sound into an electrical signal. You can use any transducer depending upon the case.
Once you have an analog electrical signal, we pass it through an operational amplifier (OpAmp) to condition the analog signal. Basically, we amplify the signal. Or limit it to protect the
next stages.
The anti-aliasing filter is an essential step in the conversion of analog to a digital signal. It is a
low-pass filter. Meaning, it allows frequencies up to a certain threshold to pass. It attenuates
all frequencies above this threshold. These unwanted frequencies make it difficult to sample
an analog signal.
The next stage is a simple analog-to-digital converter (ADC). This unit takes in analog signals
and outputs a stream of binary digits.
The heart of the system is the digital signal processor. These days we use CMOS chips (even
ULSI) to make digital signal processors. In fact, modern processors, have high-speed, high
data throughputs, and dedicated instruction sets.
The next stages are sort of the opposite of the stages preceding the digital signal processor.
The digital-to-analog converter does what its name implies. It’s necessary for the slew rate of
the DAC to match the acquisition rate of the ADC.
The smoothing filter is another low-pass filter that smoothes the output by removing
unwanted high-frequency components.
The last op-amp is just an amplifier.
The output transducer is a speaker in our case. You can use anything else according to your
requirements.
Sampling
• Most discrete-time signals come from sampling a continuoustime signal, such as speech and audio signals, radar and sonar
data, and seismic and biological signals.
• The process of converting these signals into digital form is
called analog-to-digital (A/D) conversion.
• The reverse process of reconstructing an analog signal from its
samples is known as digital-to-analog (D/A) conversion.
Analog-to-Digital Conversion
• An A/D converter transforms an analog signal into a digital
sequence.
• The input to the A/D converter, xa(t), is a real-valued function of a
continuous variable, t .
• Thus, for each value of t, the function xa(t) may be any real number.
• The output of the A/D is a bit stream that corresponds to a discretetime sequence, x(n), with an amplitude that is quantized, for each
value of n, to one of a finite number of possible values.
Sampler (C/D converter)
Quantizer & Encoder
• Because the samples xa(nTs) have a continuous
range of possible amplitudes, the second
component of the A/D converter is the quantizer,
which maps the continuous amplitude into a
discrete set of amplitudes (where T=Ts).
• For a uniform quantizer, the quantization process
is defined by the number of bits and the
quantization interval Δ.
• The last component is the encoder, which takes
the digital signal x(̂ n) and produces a sequence of
binary codewords.
Periodic Sampling
• First, the continuous-time signal is multiplied by a periodic sequence of
impulses,
to form the sampled signal
• Then, the sampled signal is converted into a discrete-time signal by
mapping the impulses that are spaced in time by Ts into a sequence x(n)
where the sample values are indexed by the integer variable n:
• Thus, X(ejω) is a frequency-scaled version of Xs(jΩ) , with the scaling
defined by
ω= ΩTs
• This scaling, which makes X(ejω) periodic with a period of 2π , is a
consequence of the time-scaling that occurs when xs(t ) is converted to
x(n).
Problems