## 1 Introduction  
The observation of spectral lines can be used to accurately study the physical properties of molecular clouds in the formation area of massive stars, e.g., the temperature, density, radiation field, and astrochemical processes [1, 2, 3, 4, 5]. Therefore, the observation of spectral lines is critical in radio astronomy. During radio-astronomical observations, digital backends are generally required to have a wide bandwidth and high-frequency resolution [6, 7, 8, 9]. To trace smaller-scale structures and more detailed dynamic characteristics, spectral line observations with high-frequency resolutions are conducted to extract information from the target celestial bodies. A frequency resolution of several hertz or 1 Hz is required in numerous high-precision spectral line observations. For example, during the Zeeman effect observation, if a neutral HI is observed, the frequency shift generated by each micro-Gauss magnetic field is 2.8 Hz. This requires a digital backend with a frequency resolution that should be higher than 2.8 Hz [10]. In addition, when observing the spectral lines, especially in the high-frequency band, the broadband receivers can simultaneously receive more information to improve the observation efficiency. The requirements of the wideband and high-frequency resolution make it difficult to obtain the digital backends for radio-astronomical observations. At present, most signal processing techniques in the digital backends, such as spectral analysis, are implemented with a field programmable gate array (FPGA). Alternatively, according to the task requirements, signal pre-processing is first achieved in the FPGA, and further processing is performed by a computer with graphics processing units [11; 12; 13]. The maximum number of points of the available fast Fourier transform (FFT) intellectual property (IP) cores in the FPGAs used for the spectral analysis is 65536. If a resolution of 1 Hz is required for a signal with a 1 MHz bandwidth, a 106-point plain FFT is required, which necessitates a specialized design with a significant workload. The operating frequency of an actual FPGA is generally several hundred MHz. This indicates that a signal with an ultra-wide bandwidth greater than several hundred MHz cannot be directly processed by a simple FFT implemented on FPGA. Therefore, the direct application of a large FFT IP core in an FPGA to the ultra-large-scale FFT processing of wideband signals cannot be achieved to obtain an ultra-high-frequency resolution. Two methods are mainly used in radio astronomy to achieve this objective. A potential method involves the use of two-stage FFTs [14]. A commonly used method involves the combination of a polyphase filter bank (PFB) (sometimes called a WOLA filter bank) and an FFT or cascaded PFB structure [15; 16; 17; 18; 19; 20]. For example, the combination regime is used as a spectrum analyser for the search for extra-terrestrial intelligence (SETI). It has a frequency resolution of 91 kHz when the processing bandwidth is 1.5 GHz, and a maximum frequency of 1 Hz when the processing bandwidth is 150 MHz [7; 19; 21]. In comparison with the two-stage FFT method, the aforementioned combined structure improves the frequency response characteristics for the first-stage channelised processing and the computational efficiency. This paper proposes a modified combination of the PFB + FFT to obtain improved flexibility with a more simplified design, and to conserve hardware resources in the FPGA. In addition, this paper proposes a IWOLA + WOLA algorithm to demonstrate an improved performance. In this paper, a spectral analysis algorithm with an ultra-high-frequency resolution is first proposed for broadband real-valued signals. This algorithm uses an improved WOLA (IWOLA) [22] channelised filter bank to channelise the broadband signals preliminarily, and decimates the output data from the IWOLA filter bank appropriately.