Digital Signal Processing Reference
In-Depth Information
Once an analog signal has been sampled, the resulting number sequence is divorced from real
time, and the only way to reconstruct the original signal properly is to send the samples to a DAC at
the original sampling rate. Thus it is critical to know what the original sampling rate was for purposes of
reconstruction as a real time signal.
When dealing with just the sequence itself, it is natural to speak of the frequency components
relative to the Nyquist limit.
• It is important to understand the concept of normalized frequency since the behavior of sequences
in digital filters is based not on original signal frequency (for that is not ascertainable from any
information contained in the sequence itself ), but on normalized frequency.
Not only is it standard in digital signal processing to express frequencies as a fraction of the
Nyquist limit, it is also typical to express normalized frequencies in radians. For example, letting
k
=0in
the expression
W
n
n
=[
exp
(j
2
πk/N)
]
yields the complex exponential with zero frequency, which is composed of a cosine of constant amplitude
1.0 and a sine of constant amplitude 0.0. If
N
is even, letting
k
=
N/
2 yields the net radian argument of
π
(180 degrees) and the complex exponential series
W
n
n
1
)
n
=[
exp
(jπ)
]
=
(
−
which is a cosine wave at the Nyquist limit frequency.
Thus we see that radian argument 0 generates a complex exponential with frequency 0, the radian
argument
π
generates the Nyquist limit frequency, and radian arguments between 0 and
π
generate
proportional frequencies therebetween. For example, the radian argument
π/
2 yields the complex expo-
nential having a frequency one-half that of the Nyquist limit, that is to say, one cycle every four samples.
The radian argument
π/
4 yields a frequency one-quarter of the Nyquist limit or one cycle every eight
samples, and so forth.
Hence the normalized frequency 1.0 (the Nyquist limit) may be taken as a short form of the radian
argument 1.0 times
π
, the normalized frequency of 0.5 represents the radian argument
π/
2, and so forth.
• Radian arguments between 0 and
π
radians correspond to normalized frequencies between 0 and
1, i.e., between DC and the Nyquist limit frequency (half the sampling rate).
Example 3.7.
A sequence of length eight samples has within it one cycle of a sine wave. What is the
normalized frequency of the sine wave for the given sequence length?
A single-cycle length of two samples represents the Nyquist limit frequency for any sequence. For a
length-eight sequence, a single cycle sinusoid would therefore be at one-quarter of the Nyquist frequency.
Since Nyquist is represented by the radian argument
π
, the correct radian frequency is
π/
4 or 0.25
π
, and
the normalized frequency is 0.25.
Example 3.8.
A sequence of length nine samples has within it 1.77 cycles of a cosine wave. What is
the normalized frequency of the cosine wave?
