This document discusses frequency concepts in continuous and discrete time signals. For continuous time signals, frequency is defined as cycles per second and relates to the periodic nature of sinusoidal signals. Discrete time signals are periodic only if the frequency is a rational number. The fundamental period is the smallest value that makes the signal periodic. As frequency increases for both continuous and discrete signals, the number of oscillations increases but the period decreases.

1. Frequency Concept in Continuous
and Discrete Time Signals

2. Introduction
 The concept of frequency is familiar to the students in engineering and
sciences.
 In radio receivers and spectral filters, frequency is the basic term.
 We also know the relation between frequency and time interrelation.
 Thus we should expect that the nature of time (Continuous or Discrete)
would affect the nature of frequency, accordingly.
 We will try to observe the frequency variations over continuous time
signals and relate the same variations over discrete time signals.

3. Continuous Time Signal
• Given an continuous time signal
𝑥 𝑎 𝑡 = 𝐴 × cos(Ω𝑡 + ϴ) with ‘t’ varying from – ∞to ∞
• The signal is completely characterized by three parameters
A- ‘Amplitude’
Ω- The frequency in radians per second
Θ- Phase in radians
• The given signal can be also represented in the form of frequency ‘F’ as
𝑥 𝑎 𝑡 = 𝐴 × cos(2 ∗ 𝑝𝑖 ∗ 𝐹 ∗ 𝑡 + ϴ)
F=
1
𝑇
cycles per second
T= period of sinusoid

5. Properties
 For every fixed value of frequency F, 𝑥 𝑎 𝑡 is periodic with period T,
where 𝑇 =
1
𝐹
is the fundamental period then 𝑥 𝑎 𝑡 + 𝑇 = 𝑥 𝑎 𝑡
𝑥 𝑎 𝑡 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ (𝑡 + 𝑇) + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + 2 ∗ 𝑝𝑖 ∗ 𝐹 ∗ 𝑇 + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + 2 ∗ 𝑝𝑖 ∗ 𝐹 ∗
1
𝐹
+ ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + 2 ∗ 𝑝𝑖 + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + 360 + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝐴 × cos(2 ∗ 𝜋 ∗ 𝐹 ∗ 𝑡 + ϴ)
𝑥 𝑎 𝑡 + 𝑇 = 𝑥 𝑎 𝑡
 Period is defined as the amount of time (expressed in seconds) required to
complete one full cycle.

6. Contd..
• Continuous time sinusoids with distinct frequencies are themselves
different.
• Increase in the frequency F results in the increase in the rate of oscillation
of signals. Also more periods are included in the given time interval.
• As the frequency increases the number of oscillations per second increases
and also the time period decreases.
• The one that is having the lowest time period is the one with high rate of
oscillations.

7. 0 0.5 1 1.5 2 2.5 3
x 10
-3
-5
0
5
Signal of 2KHz
0 0.5 1 1.5 2 2.5 3
x 10
-3
-5
0
5
Signal of 4KHz
0 0.5 1 1.5 2 2.5 3
x 10
-3
-5
0
5
Signal of 6KHz

8. Phasor Representation
 𝑥 𝑎 𝑡 = 𝐴 × cos(Ω𝑡 + ϴ)
 We know that
𝐶𝑜𝑠 𝜙 =
𝑒 𝑗𝜙 + 𝑒−𝑗𝜙
2
 Thus
A𝐶𝑜𝑠 Ω𝑡 + ϴ = A
𝑒 𝑗(Ω𝑡+ϴ)+𝑒−𝑗(Ω𝑡+ϴ)
2
 As the time progresses the phasors rotate in
the opposite directions with angular
frequencies ±Ω radians/second.
 As the angular frequencies considered
negative and positive values, we can have
positive frequency and negative frequency,
there is no limit for frequency.
 This frequency can be varied from -∞ to ∞

9. Discrete Time Sinusoidal Signals
• A discrete time sinusoidal signal may be expressed as
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 (𝑤𝑛 + 𝜃) −∞ < 𝑛 < ∞
Where n is the integer variable called sample number
– A amplitude of the sinusoid
– 𝑤 is frequency in radians per sample
– 𝜃 is phase in radians
• The above equation can be rewritten as
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑛 + 𝜃
The frequency ‘f’ has dimensions of cycles per sample.

10. Properties
• Discrete time sinusoids are periodic only if its frequency f is a rational
number.
𝑥 𝑛 + 𝑁 = 𝑥(𝑛) the smallest value of N is called fundamental period.
Given Signal
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑛 + 𝜃
𝑥 𝑛 + 𝑁 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ (𝑛 + 𝑁) + 𝜃
𝑥 𝑛 + 𝑁 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑛 + 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑁) + 𝜃
𝑥 𝑛 + 𝑁 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑛 + 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑁 + 𝜃

11. Contd..
𝑥 𝑛 + 𝑁 = 𝐴 × 𝐶𝑜𝑠 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑛 + 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑁 + 𝜃
• The system will be periodic only when
2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑁 = 2 ∗ 𝜋 ∗ 𝑘 where k is an integer.
• So the condition to become periodic is 𝑓 =
𝑘
𝑁
• Remember k, N are integers.
• Discrete time sinusoids are periodic only if its frequency f is a rational
number, also the smallest value of N is called fundamental period.
•

12. Contd..
Example
• To determine the fundamental period of sinusoid k and N should be
relatively prime.
• Given
• 𝑓1 =
21
40
=
𝑘
𝑁
𝑓2 =
20
40
• The fundamental period in case of f1 is ‘40’
• While in case of 𝑓2??
• 𝑓2 =
20
40
=
1
2
hence relatively prime and the fundamental period is ‘2’.
• If both k and N are integers then only you should consider it as
fundamental period.

13. Identify the fundamental period (N) of the
signals
• 𝑥 𝑛 = cos 0.25𝜋𝑛
• 𝑥 𝑛 = cos
5𝜋
4
𝑛 + sin
3𝜋
6
𝑛

14. Hints
• 𝑥 𝑛 = cos 0.25𝜋𝑛
• 𝑥 𝑛 = cos
5𝜋
4
𝑛 + sin
3𝜋
6
𝑛
• Relate the signals with cos(2𝜋𝑓𝑛)
• Identify f, relate with (k/N).
• Deduce till there is no common factors between
numerator and denominator and identify N
• In case if the same equation results two sinusoids,
identify N separately and find the LCM of both
N’s and that is the fundamental period of the
entire signal.

15. Answers
• 𝑥 𝑛 = cos 0.25𝜋𝑛 , N=8;
• 𝑥 𝑛 = cos
5𝜋
4
𝑛 + sin
3𝜋
6
𝑛 N1=4, N2=8
• So LCM is 8.

16. Contd..
 Discrete time sinusoids whose frequency separated by an integer multiple
of 2π are identical.
Let us take the example
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤0 ∗ 𝑛 + 𝜃 where 𝑤0is the frequency per sample.
𝑥1 𝑛 = 𝐴 × 𝐶𝑜𝑠 (𝑤0 + 2π) ∗ 𝑛 + 𝜃 ………(1)
𝑥1 𝑛 = 𝐴 × 𝐶𝑜𝑠 (𝑤0 ∗ 𝑛 + 2π ∗ 𝑛 + 𝜃 , where n is the integer.
𝑥1 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤0 ∗ 𝑛 + 𝜃
Let us write a conclusion from equation (1) as
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤 𝑘 ∗ 𝑛 + 𝜃 where k from 0,1, 2…are identical.

17. Contd..
• 𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤 𝑘 ∗ 𝑛 + 𝜃 where k from 0,1, 2…are identical.
𝑤 𝑘 = (𝑤0+2𝑘𝜋)𝑛 with −𝜋 ≤ 𝑤0 ≤ 𝜋
 The highest rate of oscillation in a discrete time sinusoid is attained when
𝑤 is π (- π)
Let us take a sinusoid 𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤0 ∗ 𝑛
and 𝑤0 values as
𝜋
8
,
𝜋
4
,
𝜋
2
, 𝜋,
5𝜋
4
,
3𝜋
2
then we know 𝑤0 = 2𝜋𝑓
Then 𝑓 =
1
16
,
1
8
,
1
4
,
1
2
,
5
8
,
3
4
also the Periods are 16, 8, 4, 2, 8, 4
• So as the frequency increases the period decreases till π and again increase
and decreases.

18. How to Plot them???
What happens when 𝒘 𝟎 =2π???

19. Contd..
• Let us take a signal
𝑥 𝑛 = 𝐴 × 𝐶𝑜𝑠 𝑤 ∗ 𝑛 + 𝜃
• The same can be represented as exponentials
as
A
𝑒 𝑗(𝑤𝑛+𝜃)
+ 𝑒−𝑗(𝑤𝑛+𝜃)
2
Then as explained in the case of continuous we
will have negative and positive frequencies

20. Harmonically related complex
exponentials
 Harmonically related complex exponentials are the sets of periodic
complex exponential with fundamental frequencies those are
multiples of single positive frequency.
 The basic signals for continuous time harmonically related
exponentials are
 𝑠 𝑘 𝑡 = 𝑒 𝑗𝑘Ω0 𝑡 = 𝑒 𝑗𝑘2𝜋𝐹0 𝑡 𝑘 = ±0, ±1. .
 Similarly discrete time harmonically related complex exponentials by
𝑠 𝑘 𝑛 = 𝑒 𝑗𝑘2𝜋𝐹0 𝑛 𝑘 = ±0, ±1. .
 The properties (periodicity..etc) that are applicable for sinusoidal
signals holds also good for these complex exponentials.