Signal

Signal: A signal is defined as any physical quantity that varies with time, space, or any other
independent variable or variables. Mathematically, we describe a signal as a function of one or
more independent variables. For example, the functions

( )

( )

( ) ( ) ( ) ( )

where t is independent variable and ( ), ( ), ( ) are dependent variable.

Another example, consider the function

( )

The function describes a signal of two independent variables x and y that could represent the two
spatial coordinates in a plan.

Basically signal is two type _____

(1). Analog signal

(2). Digital signal

What Are Analog Signals?

Analog signals were used in many systems to produce signals to carry
information. These signals are continuous in both values and time.
What Are Digital Signals?
Unlike analog signals, digital signals are not continuous, but signals are discrete in
value and time. These signals are represented by binary numbers and consist of
different voltage values.
  Define periodic and aperiodic signal.

A signal completes a pattern within a measurable time frame and

repeats that pattern over subsequent identical periods. This signal is called periodic
signal.

value value

Time Time
value value

Time Time

a. Periodic Signal b. Aperiodic Signal

Figure 02. Periodic & Aperiodic Signal

 A signal, it changes without exhibiting a pattern or cycle that repeats
over time, called aperiodic signal.

Fig 2.2(d): composite signal with several harmonics

  Define period and frequency. What is the relationship between period and
frequency?

The amount of time a signal needs to complete one cycle, in seconds, is

called period.

The number of periods in one second is called frequency.

Frequency and period are inverses of each other.

and

Where, = frequency in Hz, and

= period in second

Amplitude Six periods in 1s Frequency = 6 Hz

1s

Time
𝑇
Period= 6s

Figure 05. The concept of period and frequency

 What do you mean by phase?

The phase describes the position of the waveform relative to the time zero.
If we think of the wave as something that can be shifted backward or forward along the time
axis, phase describes the amount of that shift.

Phase is measured in degrees or radians. As shown in figure 06,

 Figure 06. Relationships between different phases

A phase shift of 0 corresponds to no shift of a period; a phase shift of

90 corresponds to a shift of one quarter of a period; a phase shift of 180 corresponds to a shift
of one-half of a period.

 What is time domain and frequency domain? Give example.

The time domain is instantaneous amplitude with respect to time. As shown

in figure 07,

5
Time

1s

a. A signal with frequency 6

Figure 07. Time domain

The frequency domain is peak amplitude with respect to frequency. As

shown in figure 08,
 5

Frequency
6𝑓 a. A signal with frequency 6

Figure 08. Frequency domain

 What is composite signal?

A signal, in which more than one frequency exists, called composite signal.

When we change one or more characteristics of a signal-frequency signal, it

becomes a composite signal made of many frequencies.

 What is the spectrum of a signal?

The description of a signal using the frequency domain and containing all its
components is called frequency spectrum of that signal. For example, figure 09 shows the
frequency spectrum of a square wave;

…….

a. Frequency spectrum of a square wave

Figure 09. Frequency spectrum

  What is bandwidth?

The range of frequencies that a medium can pass is called its bandwidth.
The bandwidth is a property of medium. It is the difference between the highest and the lowest
frequencies that the medium can satisfactorily pass.

Where, = bandwidth
= highest frequency, and
= lowest frequency

Example:
If a periodic signal is decomposed into five sine waves with frequencies of
100, 300, 500, 700, and 900 Hz, what is the bandwidth? Draw the spectrum. Assuming all
components have a maximum amplitude of 10 V.

Solution:
Let be the highest frequency, the lowest frequency, and the
bandwidth. Then

The spectrum has only five spikes, at 100, 300, 500, 700 and 900. As shown
in figure 10;

amplitude

10 V

100 300 500 700 900 Frequency

Bandwidth = 900 – 100 = 800 Hz

Figure 10. Spectrum

 What is the bit interval and bit rate?

The bit interval is the time required to send one single bit.
 The bit rate is the number of bit intervals per second, usually expressed in
bit per second (bps).

As shown in figure 11;

amplitude
1s = 8 bit intervals
Bit rate = 8 bps

1 2 3 4 5 6 7 8

……….

Time
Bit interval

Figure 11. Bit rate and Bit interval

Example:
A digital signal has a bit rate of 2000 bps. What is duration of each bit(bit
interval)?

Solution:
The bit interval is the inverse of the bit rate.

Bit interval =
 What do you mean by low-pass and band-pass channel?

A low-pass channel has a bandwidth with frequencies between 0 and .

The lower limit is 0, the upper limit can be any frequency (including infinity).

A band-pass channel has a bandwidth with frequencies between and .

Figure 12. shows the bandwidth of a low-pass channel and band-pass channel.

Amplitude

0 𝑓
Low-pass channel
Frequency
Amplitude

𝑓 𝑓 Frequency
Band-pass channel
Figure 12. Low-pass and Band-pass
Multichannel and Multidimensional Signals
Stationary and Non-stationary Signal
Stationary signals consist of spectral components that do not change in time
 All spectral components exist at all time
 No need to know any time information
 FT works well for stationary signals

 However, most of carrying signals are non-stationary, so we need to know

whether and also when an incident was happened.

Non-stationary signals consists of time varying spectral components

Symmetric (Even) and Antisymmetric (Odd) signals
The time interval T between successive samples is called the sampling period or sample interval
and its reciprocal is called the sampling rate (samples per second) or the sampling
frequency (hertz).

To establish this relationship between analog and digital signal

Consider an analog signal sinusoidal signal of the form
( ) ( )

Which, when sampled periodically at a rate samples per second, yields

( ) ( ) ( )
( )

Example 1: Consider the analog signal

( )

(a) Determine the minimum sampling rate required to avoid aliasing.

(b) Suppose that the signal is sampled at the rate . What is the discrete-time signal
obtained after sampling?

(c) Suppose that the signal is sampled at the rate . W hat is the discretetime signal
obtained after sampling?
Example 2: Consider the analog signal
( )
What is the Nvquist rate for this signal?

Solution The frequencies present in the signal above are

Thus ,

The Nvquist rale is . Hence

Consider the analog signal

( ) 6

(a) What is the Nvquist rate for this signal?

(b) Assume now that we sample this signal using a sampling rate Fs = 5000 samples/s. What is
the discrete-time signal obtained after sampling?
(c) What is the analog signal y„(r) we can reconstruct from the samples if we use ideal
interpolation?

(b) Here